The Coloring of Graphs

Annals of Mathematics

The Coloring of GraphsAuthor(s): Hassler WhitneySource: Annals of Mathematics, Second Series, Vol. 33, No. 4 (Oct., 1932), pp. 688-718Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1968214 .

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THE COLORING OF GRAPHS.'

BY HASSLER WHITNEY.2

Introduction. In another paper, L,8 the author has given a proof of a formula for M(2), the number of ways of coloring a graph in A colors, due to Birkhoff. The numbers mij, in terms of which M(2) is expressed, are here studied in detail; a method of calculating them is given.

In Part I elementary properties of the m2j and the mi and of M(2) are given. The expression for the rnij in terms of the broken circuits of the graph gives rise to sets of numbers ai and 8s, in terms of which M(R) may also be expressed. In Part II it is shown that mij may be expressed as a polynomial in numbers N1, AT2, * * *, numbers of non-separable subgraphs, which are easier to count in a given graph than are the mij. Part III is devoted to the theory of an algebraic transformation. This is made use of in Part IV to show that if we know the linear terms fi, of the above polynomial, the whole polynomial may be calculated. Finally, the polynomials for the fij, mij and mi are calculated for a few values of the subscripts.

I Received February 16, 1932.-Presented to the American Mathematical Society, Oct. 25, 1930. This is, in revised form, the author's Harvard thesis. The section on recursion formulas has been left out; Part V has been added. An outline of the paper will be found in the Proceedings of the National Academy of Sciences, 17 (1931), pp. 122-125.

2National Research Fellow. 3References will be made to the following papers by the author.

I. A logical expansion in mathematics (L); Bulletin of the American Mathematical Society, 38 (1932), pp. 572-579.

II. Non-separable and planar graphs (N); Transactions of the American Mathematical Society, 34 (1932), pp. 339-362.

III. Characteristic functions and the algebra of logic (C. F.); to appear in the Ann. of Math. IV. Congruent graphs and the connectivity of graphs (C. G.), American Journal of Mathe-

matics, 54 (1932), pp. 150-168. The notations in N will be made use of. We recall the following. The rank R and

nullity N of a graph G containing V vertices and E arcs which is in P connected pieces are given by the equations R=V-P, N = E -R E-V + P. A subgraph H of G is determined by naming a subset of the arcs of G. A graph is non-separable if it is

connected and cannot be broken at a single vertex into two pieces. The components of a graph are the non-separable parts of the graph. The sum of two graphs G1 and G2 is

the graph G1 + G2 containing the arcs and vertices of both graphs, provided they had no

vertices in common. If they have common vertices, we consider these vertices as distinct in G1 + G2.

A graph is colored by assigning to each vertex a color in such a way that no two

vertices which are joined by an arc are of the same color. 688



THE COLORING OF GRAPHS. 689

With reference to the four color map problem, we mention the following theorem. If G is a planar graph and G' is a dual of G, and mij and miy are their numbers, then

.4 mVj mR-j,N-Z.

This follows immediately from the definition of dual graphs in N. Con- sider the class of graphs G with numbers m2j for which the above numbers mij are also the numbers for a graph G'; this class then includes all planar graphs (see N, Theorem 29). Hence a proposition which implies the four color map theorem is the following: For any graph G of the above class, j(-1)i+imij4v-i>0. (We must, however, exclude graphs ij for which anMi 4 0; see ? 3.) This proposition is stronger than the four color map theorem; for there are graphs in the above class which are not planar.5

I. THE POLYNOMIAL M(i); THE NUMBERS a, AND 8f. 1. Some properties of the mij. We showed in L that if there were

mij subgraphs of a of rank i, nullity j, then the number M(2) of ways of coloring a is6 (1.1) M(i) (1)i+j Mj I-i = mi A V-i,

ij i (1.2) mj ; )i+jmi.

If GC and G2 are the two graphs mentioned at the end of L (T21 and T32 in Fig. 2), whose arcs are a(ab), fi(ac), r(bc) and a(ab), 8i(ac), y(bc), d (bOd), e (c d) respectively, we find for their numbers mij those given in the following tables.

nu lity __ 1 22 oil 0 1 0 0

0I 1 0 G2 1 5 0 0 ,~~~ 1 3 0 ~~~2 10 2 0

2 3 1 3 8 15 1

From these tables we immediately find their numbers mi as given at the end of L. For instance, in GI, in2, the coefficient of A, is m2 - m21 = 3 -1 = 2.

4The formula of Theorem 8 of the Proceedings paper should be corrected. 5An example was recently found (May 13) by R. M. Foster: G contains the arcs ab,

ac, bc, ad, de, ec, bf, f g, gc, eg, dg, fe. The corresponding graph G' is this same graph.

6Throughout this paper, sums are carried out over all nonhzero terms unless otherwise stated. We put always () = 0 if q<0 or q>p.



690 H. WHITNEY.

Moo is always 1, for there is always exactly one subgraph of any graph (even of the null graph, containing no arcs or vertices), of rank 0, nullity 0, namely, that containing no arcs. If the graph contains no 1-circuits, each arc is of rank 1, nullity 0, and each subgraph of rank 1, nullity 0 contains just one arc. Thus m1o -- E. We shall always assume in the future that there are no 1-circuits in the graph, unless expressly mentioned. Then also of course moi = 0. In any case, mi1 + mol = E. There is just one subgraph of G of rank R, nullity N, namely G itself, and MRN = 1. Thus

(1.3) moo = mRN = 1, m10 = E (or m1o+ mol = E). Of course (1.4) mi=- 0 if i<O, j<0, i>R or j>N.

There is a fundamental set of relations between the mi1:

(1.5) mo l+ mi--, ? + mos =

for the sum on the left is the total number of subgraphs of G of i arcs. We shall see in Part IV that there is no other linear, or even polynomial, equality connecting the mi.

2. Some properties of the mi. We can derive immediately a number of properties of the coefficients mi of M(R) from the interpretation of (- 1)i mi in terms of the broken circuits of a given in L. Let us show that ( l) miR> 0 . We drop out all the arcs of 0 we can without dis- connecting any vertices that were formerly connected. The resulting graph H is a forest' of rank R. Order the arcs of G in a definite manner, choosing the arcs not in H first. Every circuit in a contains an arc not in H, and therefore every broken circuit contains an arc not in H. Thus H is a subgraph of 0 of R arcs not containing all the arcs of any broken circuit, and hence (- 1)" mR> 0. Taking any subgraph of H, we see that

(2.1) ( )i mi >O i O. 1, .. * *,B. Also (2.2) mi O. i<0 or i>R

from eq. (1.4). Hence M(A) has no constant term, as it would be my; but V>1? always. If G is connected, 1 = V-1, and the coefficient of A in M(R) is mv + ?O

3. Some properties of M(2). As M(R) has no constant term, M(0) O. This is obvious, as no graph can be colored in no colors.

Let us find M(1). AU1- A gp )i+t mv0 i j

7 A graph of nu~ty 0.




or, if we sum diagonally,

M(1) - J(1 )i Z i-ksk = f (-1 E

iki by eq. (1.5). Hence

M(l) = (1 - 1)E 0

if E> 0. If E 0, 21(1) = moo= 1. Thus a graph containing an arc cannot be colored in one color; if it contains no arc, there is one way of coloring it in one color.

Let us find M(A) for a circuit of E arcs. Every subgraph of the circuit is of nullity 0 except the whole graph, which is of nullity 1. Hence

Mio E(*), 0 < i < E-1, ME-1,1 =1 .

Therefore M( E

(31)M(2) = (_ )i E

)) vE+( I)EXV-(E-1)

=(2 1)E+ (1)E (2 1),

remembering that V - E. We can also prove this by noting that there is just one broken circuit, which contains E- 1 arcs. In particular, M(2) = 1?+((- I)E, which is 2 if E is even and 0 if E is odd.

Suppose the graph G contained a 1-circuit aa. Then in any coloring of the graph, a must be colored in a different color from itself, which is not possible. Consequently M(O) = M(1) - ... 0, and thus M(%) 0. We can prove this from L, eq. (11), by pairing off terms, so that the two terms of a pair are the same except that one contains the factor Aaa while the other does not. The two terms of each pair cancel.

4. The numbers ai and di. We shall illustrate these numbers by means of an example, using the graph G2 of ? 1. Its circuits are aflr, y GE, al 6E; its broken circuits are afi, r 6, aldd. (-1)i m is the number of subgraphs of U2 of i arcs not containing both a and a and not containing both y and d. For under these conditions, it does not contain a, fi and d either. Let Ri be the set of subgraphs of G2 of i arcs, let nj (a) be the number which contain the arc a, etc. Then

( 1)imi = ni(a-.fl-r.).

Expanding this as in C. F. into the second normal form, we have

(4.1) ni(1-afl-y6+aflrd) =. - (1) -ni(al) -ni(r6)+ni(aflyd).



692 H. WHITNEY.

Consider a typical term, such as nf (a fi). This is the number of subgraphs of i arcs which contain both a and f. To form such a subgraph, we must pick out the two arcs a and i; we have left then i -2 arcs which

we can pick out from the remaining 5-2 = 3 arcs of G( in (5)

ways. This then is the value of the term. The sign of the term is (- 1)1, where I is the number of broken circuits combined in forming the term; in the above case, 1. In general, if the term is formed from I broken circuits, and if it contains k distinct arcs, then the value of the term is

( E)l(-k) Definition. Let aki be the number of sets of I broken circuits which

together contain k arcs. Put

(4.2) ak = V(-1Y ad.

Then

(-I)~~ =i _ (-1) (ik) a kk = (i-k) atc.

We note that ak is just the sum of all the coefficients in eq. (4.1) which contain k arcs, and thus ak is the ak defined in C. F. at the end of ? 10.

If we expand eq. (4.1) into the third normal form, we find

(4.3) (-'* = ni ay a ? +? 18J- aT6? f?r fidd,86)

We define fik as being the sum of the coefficients of all terms in this equation containing k arcs, again as in C. F.

With the help of C. F., eq.'s (22) and (23), we can express any of the sets mi, ai, fi in terms of any other of the sets. The relations ares

8 The formulas not already proved may be proved from the others by use of the relation

(-, q (pq)I if pq;

see Netto, Lehrbuch der Combinatorik, Leipzig, 1927, ? 158, (27). For instance, to prove the formula for as in terms of mk, put

(-V a,- _1 I-k M) Ek=z -k ) _lkz(E-1 at

-A; z i-k( i) k (k-I I

k E ik-

Replacing k by k +1, we find

(- (-k E- ) k

z (-1)ic (1 0) = (-1)'ai.

Thus as = as, as required.




E -kC)- -k,

(4,4) (-)i ai =

E- (k i

k ( a) ( )i i- ( 1)Ez (Eki) mk = (k)

The numbers for G2 are

ao - C1, a2 2, a4 1;

82 =4, A 4, ~4 1

M- 1, ml 5, m2 8, m3 -4, m4 0.

The first non-zero Aii is fiE-RR NiN. Remembering that mk = 0, k>R, we find from the last equation above

(4.5) fiN - (- 1) mR,

which is the last non-zero coefficient in M(2); the coefficient of i, if the graph is connected.

In terms of the av, we have

(4.6) M(2) - VE (l)i a,(- I)E-i

For, calling this expression (A),

= ( VE ( l)E ai (1 A)F-i

_ V E (l)E ai (l)k A(E )k i ~~~k

by the binomial formula, = A V-E+k ( l)-k (E-i) vEi,

or, replacing k by E- k,

T(A) _ REV-k(-J)k2: (E k ai - ,Smk2V k M(i).

k

We shall see by examples in Part V that the ai are given by simpler formulas in terms of the numbers of non-separable subgraphs of 0 than are the mi.

II. Mii IN TERMS OF NON-SEPARABLE SUBGRAPHS. 5. The numbers N1, N2, *... To find mij, we count all the sub-

graphs of 0 of rank i, nullity j. In this part we shall show that it is



694 H. WHITNEY.

not necessary to count all the subgraphs; we need count only the non- separable subgraphs. We say a subgraph is non-separable if it becomes so when any isolated vertices there may be are dropped out. This operation alters neither the rank nor the nullity of the subgraph.

We note that Theorem 7 A holds even if we allow the graphs to contain 1-circuits. If G' is G except for its moi 1-circuits, then obviously

(5.1) my = ~(mo1) me j-ke

Let us say two graphs are of the same type if they are congruent. Make a list of all types of non-separable graphs; call these types T1, T2, T, .... There are of course but a finite number of types of graphs with a given number of arcs. Let Ni be the number of subgraphs of G of type Ts, i = 1, 2, .. The fundamental theorem of this part is that mij for any graph G is given by a unique polynomial in the numbers N1, N2, ... for G. When we speak of a polynomial, we mean always a polynomial in a finite number of variables. Thus we do not allow expressions such as N1 + N2+ N +., even if, for any given graph, there are but a finite number of non-vanishing terms.

THEOREM 5 A. Any polynomial in the numbers N1, N2, * , which equals 0 for all graphs is identically 0.

Suppose the theorem false. Then there is a polynomial which is 0 for all graphs but not 0. Let P be such a polynomial containing the least possible number, say q, of variables. Let us call these variables N1, N2, v Nq, where T1, corresponding to N1, is one of the types of graphs concerned containing the least number of arcs.

Arrange the polynomial according to descending powers of N1, and say it is of degree a in this variable:

P(N1, N2, ... , Nq) = PO(N2, *, q) N+P1(X2, * , Nq) Nf' + - - - + Pa UV2 * @ In Nq) 2 o t Ox-

Take now any graph G, and form from it graphs G1, G2,.*, Ga, each being formed from the last by adding a graph of type T1, letting it have no vertices in common with the preceding graphs. As a graph of type T1 contains no subgraph of any of the types T2, ..., Tq, we have in this manner added no subgraphs of any of these types. Hence the number N1 for each graph Gi is one greater than that for the preceding graph, G2_1, while the numbers N2, * Nq are the same. P vanishes for these distinct values of N1 and the coefficients PO, P1, *., Pa are constant for these graphs, and hence these coefficients must vanish for these graphs. They vanish, in particular, for the graph G. But G was any graph, so they vanish for all graphs. But this is a contradiction. For, if q was > 1,




PO is a polynomial in a smaller number of variables, vanishing for all graphs without being identically 0, and if q was 1, P0 is a constant = 0, and P was not of degree a in N1.

THEOREM 5B. Theorem 5A holds if the words "for all graphs" are replaced by the words T~for all non-separable graphs".

Using the same notation as before, let- G be any non-separable graph. The only change necessary is to make GI, *.., G. non-separable also. Suppose first that the numbers N1, -, Nq do not include E, the number of arcs in G. Let s be the greatest number of arcs in a graph of any of the types T1, , Tq. We add a graph of type T1 to G, and then join two of its vertices to two vertices of G by distinct chains, each containing at least s/2 arcs. The resulting graph G1 is non-separable. Moreover, in adding the two chains we have added no subgraphs of any of the types T2, * , Tq. 02, ..., Ga are formed similarly. If, however, the polynomial contains the variable E, we merely add a suspended chain of s arcs each time, increasing E by s, and leaving the other Ns unchanged.

6. nij in terms of N1, N2, .... Given a graph 0, consider the set S of all its subgraphs of rank i, nullity j. Drop out any isolated vertices there may be from each subgraph. Let us list all possible graphs of rank i, nullityj which contain no isolated vertices (many may be separable), and divide them into types U1, U2, ..., Up. We say here two graphs are of the same type if they are equivalent.sa Each subgraph of S is of one of these types; say Sk, containing Sk members, is the set of subgraphs of type Uk (any or all of the Sk may be 0). Then

(6.1) =nj -sL+S2+"'+sp.

We shall show how to evaluate each Sk.

Let H be a graph of type Uk, let H1, H2, Hq be its components, named in a definite order, (there will be but one component if H is non- separable), and say Hi is of type Ti, i- 1, ..., q. (Some of these types may be the same.) We now define a set Rk as follows: Any member of Rk consists of an ordered set of non-separable subgraphs of G. We name first in 0 a subgraph H1' of type T1, next a subgraph H-2' of type T2 (which may have parts in common with BH'), .. , and finally a subgraph Hq of type Tq (which may have parts in common with any of the preceding graphs). Let nk be the number of sets of graphs in RBk. If, for instance, H consists of two arcs, its components H1 and H2 each being an arc, Rk is the set of ordered pairs of arcs in 0, the two arcs being distinct or not, and nk =E 2.

8aTwo graphs are equivalent if they have the same components.



696 H. WHITNEY.

Consider any member of Rk. It may happen that some of the subgraphs picked out of G to form this member, say the first three, form a circuit of graphs in 0. If so, we shall say that this member of Rk has the property

H, ? H2 o H3.

We are interested in those members of Rk in which no subset of the subgraphs form a circuit of graphs in G. These are

Rk (H, o He * H, o Hi s * Hq-1 ? Hq HIOH ? Hs * * * H, O H2 ? .. * *Hq)

Rk (A12 * A13 * *Aqq * A,23 *.*A2.. .q) Rk (A).

Remembering that the types T1, , Tq may not be distinct, let us pick out the distinct types T, T2, *.., TT,' from these. Say the first vi types T2 .2 TT1 are the same type as T, etc. Of course

(6.2) ti? 2?*?Tr =q. We shall now show that

(6.3) Sk flk(A).

To this end we establish a correspondence between the members of Sk and of Rk (A). Consider first any member of Rk (A); say H1', , Hq are the subgraphs of G named, where Hi' is of type Ti. Let H' be the subgraph of G containing the vertices and arcs of all these graphs. No subset of these graphs form a circuit of graphs, and hence these graphs are the components of H', by N, Theorem 17. Therefore H' is of the type Uk, and is a member of Sk. Thus to each member of Rk (A) corresponds just one member of Sk.

Consider now any member H' of Sk in G, whose components are H,*. , Hq, Hi' being of type T2. There are exactly rl! v2! * * * Tr! corresponding members of Rk(A), obtained as follows. Hi, *.., yH,' all being of the same type T1, and these being distinct in G, we name first any one of these, then any other, etc., which we can do in ri! ways, forming the first part of an ordered set of graphs of Rk (A). The next r2 subgraphs we can name in v2! ways, etc. The equation now follows. This is true in particular if all the types T1, ..., Tq are different or all the same; also if H' is non-separable, in which case, there is but one type T= T1, and r1 = 1.

Summing over k, we have

(6.4) mv = 1 - In (A).




This expression is evaluated by expanding A by the logical expansion:

(6.5 VIT1! VS .T! nk [1 (A12 + A1s + ***+ Aq-l1 (6.5) mynl(1?As A-k + A,23+ * + A12. .. .q) + (A12 A1,3 +***) **].

7. Evaluation of eq. (6.5). We shall show that a typital term of the above expression is a polynomial in the numbers N1, N2, .... This will prove that mij is a polynomial in these numbers.

Consider the term

(7. 1) nk(Aijia Ajij,... * ** AIzi...z ) nk (B B2** Bu) = nkf(B),

each Bi stating that a certain subset of the graphs H1', * , Hq form a circuit of graphs in G. This term is the number of ways of naming in G first a subgraph H1' of type T1, * , finally a subgraph Hq of type Tq, so that the relations B1, ***, B. all hold.

Divide the relations B1, * *, B. into sets as follows. Suppose two of these included the same graph; e. g. if

B1 H= H10112, B2 = H2oIH4oH5,

then B1 and B2 both include the graph H2. Say then that B1 and B2 are connected. If Bi and Bj, also Bj and Bk, are connected, say that Bi and Bk are connected. We put two relations into the same set if they are connected, otherwise, into different sets. Let C1 stand for all the relations in the first set, C2, for all those in the second set, * ,

C,' for all those in the last set. Then

(7.2) B C1 C2 ..*Cv .

Divide now the graphs H1, 112, * * Hq into sets as follows: All the graphs appearing in any relation Bi of C1 we put in the first set Q1, all appearing in a relation of C2 we put in the second set Q2, ***, all appearing in the last set C, we put in the set Qv'. There may be some graphs left over (if we are considering the first term of eq. ((.5), there are no B's, and none of the graphs Hi, Hq have been placed in sets); if so, put each of these in a set by itself, calling these sets Q1 '+, , Qv. If we define the characteristic functions

(7.3) Ci - 1, i v'+1, . .., v,

then we may write also

(7.4) B = C C2... Cv 46



698 H. WHITNEY.

where v = v' if there were no graphs left over. Now each graph Hi falls into exactly one of the sets Q,, * * , Q,.

In determining nk (B), we picked out the graphs Hz, *i , Hq from G in a definite order. Let us fix this order so that the first graphs fall into Q,, the next fall into Q2, etc. We can now say that nk (B) equals the number of ways of picking out from G first the graphs of Q, in a definite order so that Cl holds, next the graphs of Q2 in a definite order so that C2 holds, *--, finally the graphs of Q, in a definite order so that C, holds. (For values of i >v', the words "so that Ci holds" may of course be left out.) Hence if n'(Ci) is the number of ways of picking out from C graphs of the set Qi in a definite order so that Ci holds, then

(7.5) nk (B) = nk(CQ) nk(C2) ... nk(CG).

We shall show that each term of this product, such as nZ(Cj), is a homogeneous linear polynomial in the numbers N1, N2, * *, showing that nk (B) is a homogeneous polynomial of degree v in these numbers.

Let us say H1, H2, ..., Hz are the graphs in Q>. Name graphs of these types in G so that Cj holds. These subgraphs of G together form a subgraph I of G. I is non-separable. For if first Cj contains no B's, i. e. Cj = 1, then there is but a single graph H1' in Qj, and I is H1', a non-separable graph. Otherwise, I consists of a number of these graphs, various ones of which form circuits of graphs. Suppose

Cj -Bil Bis ... Bit.

Consider that part of I, I,, formed of the graphs forming the circuit of graphs corresponding to Bi. By N, Theorem 16, k1 is non-separable. Similarly, I2, ..A, IZ are non-separable. Now the relations Bi1, ..., Bi are connected. Hence there is a graph H,' (containing at least two vertices) in both Bi1 and some other relation, say Bi., i. e. in both I, and I2. Thus these two graphs form a circuit of graphs, and the two together form a non-separable graph. Continuing in this manner, we see finally that I is non-separable.

Fit the graphs H1, ..., H, of Qj together in all possible ways so that the relation Cj holds. We get various non-separable graphs, of types T1il, T2", ***, T"' say. Suppose I is of type T7". Let ji equal the number of ways of fitting first H1, next H2, ..., finally H, into I so that Cj holds and I is completely filled out. Then if there are Ni subgraphs in G of type Ti", i -1, ..., w,

(7.6) nk (Cj) = IIN1 + ?2N2 + *+NW.




To prove this, consider the correspondence between the sets of graphs counted in n' (Cj) and the subgraphs of G of types T1", . T4'. Corres- sponding to each way of picking out a set of graphs fromn G in counting nk (Cj) is a non-separable subgraph I of G of one of these types. Consider any such subgraph I of type Ti" of G. There are exactly ji ways of naming subgraphs in G properly so that these graphs together form I in G. Hence each subgraph of this type contributes an amount ji to n' (Cj); as there are Ni such subgraphs in G, they all contribute an amount lsNi, and the result is established.

As the numbers r-, r2, ..., r,, and 1' y **2 i j, depend only on the types of graphs considered and on i and j, not on the graph G, we have proved all but the statement of uniqueness in the

THEOREM 7 A. nij for any graph G is given by a unique polynomial in the numbers N1, P2, ..., containing no constant term, the coefficients being independent of the graph G.

To prove the uniqueness of the polynomial, suppose there were two such polynomials. Then their difference is a polynomial which is 0 for all graphs, and hence is identically 0, by Theorem 5A; the two polynomials are therefore identically equal.

Examples. We consider here only graphs containing no 1- or 2-circuits. m20: Each subgraph H of G of rank 2, nullity 0, consists of two distinct

arcs, H1 and H2. As H1 and IH are of the same type, ri = 2. Hence, by eq. (6.5),

mo= 2! n(H1oH2) = n(1-H1oH2).

n (1) is the number of ways of picking out first any arc, then again any arc from G. This is E2. n (H1o H2) is the number of ways of picking out first an arc H1, then an arc H2, such that the two form a circuit of graphs in G. As there are no 2-circuits in G, these arcs must be identical. This number is thus E. This gives the obvious result

M20 - 2 (E2-E) =

mil: A subgraph H of G of rank 3, nullity 1 consists either of a quadrilateral, or of a triangle E1 and an arc H2. The number of subgraphs of the first type is N31 say. The number in the second set is, as rl = 2 1, n(1 -H1H2). Now n(1) = EN21 if there are N21 triangles in G. If H1 and H2 are to form a circuit of graphs, the arc Hi must be one of the arcs of the triangle H1. Together they form a triangle l. H1 can be fitted into I in one way; the arc can be fitted in in three ways. Hence sil = 3, and n(HioH2) = 3 Ni1. Adding,

46*



700 H. WHITNEY.

m31 = N31 + (E -3)N21, a result which can be easily verified.

8. The types of non-separable subgraphs contributing to a given miy. Given two numbers i, j, we wish to know what numbers N1, N2, * * *, appear in the polynomial for mij. We are interested especially in the linear terms of the polynomial; c. f. Part IV. If the number Ns corresponding to the type T2' of graph appears in some term of the expansion (6.5) for mij, we shall say a graph of this type contributes to mv. However, it might possibly happen that the terms cancel out so that N. does not actually appear in the polynomial for mij.

As we saw in ? 7, a term such as nk (B) in eq. (6.5) is linear if all the graphs H1, H2, * * *, Hq fall into a single set Q,, i. e. if v = 1. This happens when the relations B1, * * *, B. are connected and each graph Hi falls into one of these relations, or if H is non-separable.

From the considerations of ? 6 and ? 7 we have the THEOREM 8A. The necessary and sufficient condition that a non-separable

graph I contribute to the linear terms of mij is that there exists a graph H of rank i, nullity j, with the components H1, H2, * * *, Hq (q > 1), such that if these graphs are allowed to coalesce in the proper manner, I is formed.

As an immediate application, an arc contributes to mio only. It is seen that a non-separable graph can only contribute to mij if it contributes to the linear terms of some mki, k < i, 1 ?j. (This follows also directly from the facts in Part IV.)

LEMMA A. A non-separable graph of rank i, nullity j, contributes to mij with the coefficient 1.

These are the only non-separable graphs that are counted directly in eq. (6.5).

LEMMA B. A non-separable graph of rank i - k, nullity j + k, contributes to the linear terms of mij if k >0.

Let I be such a graph. As it is of nullity >O, it contains at least two arcs. Pull out one of its arcs, and let I' be the new graph, composed of this arc, I,, and the remaining part of I, 12'. I' is of rank i - k + 1, nullity j + k- 1. Separate I2' into its components if it is separable, which does not alter its rank or nullity. If I' is still of nullity >j, we take one of its components of nullity >0, and pull out an arc as before. Con- tinuing, we obtain finally a set of non-separable graphs, the whole being of rank i, nullityj, which, if pieced together properly, form I. The lemma now follows from Theorem 8A.

LEMMA C. A non-separable graph of rank i-k -1, nullity j+k, contributes to the linear terms of mij if k > 0, 1>0.




Let I be such a graph. Add to it 1 arcs, forming the graph I' in 1+1 connected pieces. I' is of rank i -k, nullity j+k. Pull out arcs from this graph as in Lemma B until we obtain a graph of rank i, nullity j. I is formed from this graph by reconstructing I', then letting the arcs coalesce with arcs of the resulting piece.

Various graphs of rank i -k, nullity j- 1, may also contribute. If so, similar graphl of rank i - k - s, nullity j- 1, contribute.

LEMMA D. No non-separable gra~ph containing more thani +j arcs contributes. Obviously, from Theorem 8 A. LEMMA E. No non-separable graph of rank i +1, nullity j - k, contributes

to mi f 1> 0, k>0. For suppose I were such a graph which contributes to mij. Then, by

Theorem 8A, I is formed by letting the components H1, H2, * * , Hq, of which there are at least two, of a graph H of rank i, nullity j, coalesce. But then, by N, Theorem 14, I is of rank <i, a contradiction.

LEMMA F. If mij, j >0, is >0 for some graph G, then there exists a non-separable graph I of rank i, nullity j -(which contributes).

G contains a subgraph H of rank i, nullity j. If it is separable, let H1, *i*, Hq be its components. Let an arc of H1 coalesce with an arc of He; this reduces the rank of the whole by one, but leaves the nullity unchanged. Continue in this manner until we have a non-separable graph H' of rank i-q + 1, nullity j. Let ab be an arc of H'; replace it by i-X two arcs, ac and cb, c being a new vertex. This

Fig. 1. increases the rank by one but leaves the nullity un- Showing those types of changed. Continuing, we have finally the non-separable non-separable graphs graph I as required. coiltributing to the linear

These results are collected in Fig. 1. terms of a given mij

III. THE TRANSFORMATION T. 9. Definition and elementary properties of T. Let there be

a set of elements a, b, c, - , such that for any two elements a, b, there is a "sum element" a + b = b + a also in the set. (We shall later interpret these elements as graphs.) Let fi(a), i = 1, 2, ... , be a set of numbers defined for each element a, with the property that

(9.1) fi (a + b) fi (a) + fi (b) for any two elements a, b. We wish to find a transformation T from these sets of numbers to sets of numbers mi (a), i = 0, 1, 2, ... , with the properties that mo 1 and



702 H. WHITNEY.

mi (a + b) mo (a) mi (b) + ml (a) m1-, (b) + + mi (a) mo (b) (9.2) - fmk(a)mi-k (b).1

k

The following symbol will be found convenient: Rik(f) is the sum of all terms formed by multiplying together k numbers fj1, fjA, * *, Ail whose subscripts (which need not be distinct) add up to i:

(9.3) jl +j2 += * +jk - i,

and two terms are called different if their factors are arranged in different orders. Put also

(9.4) Ro(f) 1, RB(f) 0, i>0, R (f) == 0, i<k.

Thus for example R14(f) f ,

(9.5) R4 (f) f1f3 +f2f2 +f3fl == 2 fi f3 +f2,

R4 (f) AfAff2 +fJfAA +f2fifi 3 f=f2j

R4 (f) = fififif f

Now for any element a, let f, (a), f2(a), *., be the coefficients of a power series in x:

Fa (x) fl (a) x +f2 (a) x2+..*

We shall define rmi(a) as being the coefficient of xi in the power series expansion of eFa(X):

eF_(4 = 1 +Fa(X)?+ ! F1

= mo(a) + ml (a) x +m2(a)X2+* = Ma (x).

Equating coefficients determines the mi. The first few are

M 1 MI = fu, m2 f2 +2

(9.6) rn3 1 !VA+~l 13!A 13A 6 () = f3?- (f1f2+ff)? 3!f =f +f1 f2? f1,

12 12 14 m4 =f4+flf3+- -2?-2 -f~fi2+ 24Jfi

and the general term is found to be

______mi k! Xf).

See footnote 6 in ? 1, Part. I.




mi is a polynomial in flf2, *. ., fi, for i> O. If we collect terms, we find the following expression for the transformation T:

(9 ) f z k! Rt~~~~f Ekzi al! a2l . .. a,- 12

where in the second sum all terms with a1+ 2a2+ + i ai i appear. To find the inverse of T, we compare coefficients of xi in the equation

Fa (x) = log Ma(x). We find

k9.8) f A, (.t) Rt(m) = l ( 1)ss!

k k Nk = at! a2! ... ai! m 2 1

where S = +a2+ *+a?-1.

The first few fs are

12 1 (9.9) f1g = inl, f2 = m2--2-m1, f3 = m3-mim2?- j-m3. If we write T in the form

(9.10) mi =f +s + 7! Rk(f ), k 2 k.

each term of the sum contains k > 2 factors. Similarly for T-'. We have therefore the

THEOREM 9A. mi[fi], i>0, equals f[?[mn] plus a polynomial in f1, f2, ***,

fi-1 [inl, Mi2, ***, mi1], containing no constant or linear terms. We come now to the fundamental theorem. THEOREM 9B. If

fi (a + b) = fi(a) +? f(b), then

mi (a +b) z Mk(at)M Ek (b), k

and conversely. Define the power series Fa(x), Fb(x), Fa+b(X), Ma(X), Mb(x), MIa+b(X)

as we formerly defined Fa (x), Ma (x). If now fi (a + b) = fi (a) +fj (b), then

Fa+b (X) =a (X) + Fb (X). Hence

Ma+b(X) - eFa+b(X) eF (x)+Fb(X) eFa(x) eFb(X) Ma (X)Mb(X)

Equating coefficients gives the first half of the theorem. The second half is proved similarly.

10. Uniqueness of T and T-'. We shall show that T and T-i are the only transformations obeying Theorems 9A and 9B. Consider first T.



704 H. WHITNEY.

THEOREM 10A. Let U be a transformation from numbers fl, f2, ...,

to numbers ino, ml, m2, *., with the following properties. (1) mO 1. (2) Each mi, i>O, equals fi plus a polynomial in fl, f2, *, fi- con-

taining no constant or linear terms. (3) If fs (a) and fi (b) are arbitrary sets of numbers, and fi (a + b) is defined

by the equation (9.1), then the corresponding sets mj(a), mi(b), mi(a+ b) are related by the equation (9.2).

Then U is exactly T. If we take i-= 1, we find m1 must equal fl. Assuming the expressions

for ml, m2, *., mi- are uniquely determined by the above conditions, we shall show that that for mi is also.

By hypothesis, we can write

(10.1) mi = fi+IA(a,, a2, .** aCi-) f1"a f2 fail..

the sum containing no constant or linear terms; that is, in each term of 2,

(10.2) al + a2 +? * * + r?ial ? 2.

Let now fd(a), fd(b) be arbitrary sets of numbers, and define fd(a+ b) as in condition (3). If we put

(10.3) M1= mi(a+b) -mi(a) mi(b),

then by conditions (1) and (3),

(10.4) M = Ml (a) mi-,(b) + M2 (a) mi-2(b) + * + mi-l(a) m1 (b),

which contains neither mi(a) nor mi(b). But also

M fi(a+b) + ,3 A(1l, ,i-, r)f[r(a+b) ... fil(a + b)

* -2(5) 1 A (a, , ai-lg (a) * *gal (a)

-fi (b) A (fi1, ... ,fi) fI(b) **'f.'l (b) -3 A (v, * - *, ri-,) [ f (a) +fj (b)]rl ... [fj-i(a) +fj-j (b)]ri-

1 2

(10.5) = 8A (Y1 *.. * *z ~) [I ( al) fi l (a)fgrl-Ex1(b)]

xaI(i-1)f'lag'-a-b]

r-3- A(tof ,.. ... (ai) * * * (aj

x flal(a) ... fgail1 (a)f [ral (b) ... fjiril-ai (b)




where in A' all terms in which

a1 a2 *...* i- 0 or a1=Y1, a2y 2,..., ai-1 = i-1,

i. e. in which b alone or a alone appears, are left out, as these are can- celled out by terms of 1 and 12.

If in eq. (10.4) we express each mk in terms of the fl, which we can do in a unique manner, by hypothesis, the resulting expression must be identically equal to the right hand side of eq. (10.5) for all values of fi(a) and fi(b), as these values are arbitrary.

This fact is sufficient to determine A(rl, ... , Yr-1) for all sets of y's obeying eq. (10.2) (with ai replaced by Ti). For choose any such set. This is a factor of the coefficient of at least one term in (10.5). For instance, if Yp t 0, it is a factor of the coefficient of

f (a) gfir (b) ... f p (b) fgru'(b) f ++l (b) ... ffi7, (b).

For, if rp > 1, fP(a) and frP-'(b) are both present, and if = 1, some

Tq>O. and fp (a) and frj (b) are both present. The whole coefficient of this term is

A (y, * .. *, ri-1) .. ... )r Yp **(i ) p A (ri, * r * i-D).

As this must equal the corresponding coefficient in (10.4), A(ry, ... , ri-) is determined. There is thus at most a single transformation with the given properties. But T is such a transformation; hence U is T.

To prove that T-' is unique, we state a theorem corresponding to the above theorem, with the mi and fi interchanged. We assume of course that mo - 1. Now the transformation U' given in this theorem obviously has an inverse. Moreover, this inverse is seen to obey the conditions of Theorem 10A, and it is thus T. Hence U' is T-'.

11. Extension of T to the case of two subscripts. We now consider sets of numbers fj, (i, j) t (0, 0), and mij, where moo = 1. Let Rkv(f) be defined just as we defined Ri, with the condition that the first subscripts add up to i, and the second ones add up to j. Thus for example

(11.1) R22 (f ) f22 M R22 (f 2(fo f12 +?f2o fo2 +?fo f21) +?fl 142(f ) 3fifo 2?+ 6fio fol fil + 3fo21 f2o, R22(f) = 6ffol .

Letting fij(a) be the coefficient of xeyi in a power series Fa(x, y), we define mv (a) as being the coefficient of xiyi in e (xY), etc. Just as before, we arrive at the transformations T and T--:



706 H. WHITNEY.

(11.2) = 1 Jj-fPq, * pq J7Japq pgq

2qacc = p,q q pq

(11.3) fiq = ) R'(m) = y () Impq, k k =ia Jljccpq! pg

2qapq=j pq q

where S = zapq-1.

p,q Thus, for example,

1 2 m22 = f22+f1of12+f2ofo2 folf21+ 2 fif1

2 fo2 + fAf20 +fiofOifil+ 1 2O2O

(11.4) 1 122 im22 - mlo M12 - M20 M02 - mol 21 2 l2l

+ m00 in2 ? mO1 i20 + 2 mi1 o1 ml -m1, mi mO1.

As before, we prove the theorems THEOREM 1 A. mij [f,], (i, j) + (0, 0), equals fi [mV] plus a polynomial

in the numbers fpq[mpq], p < i, q < j, (p, q) + (i, j), (p, q) 1 ((0, 0), which contains no constant or linear terms.

THEOREM 11 B. If (11.5) fV (a+b) fi=(a)+fij(b), then (11.6) mij(a + b) = mpq(a) mj-p,j-q(b),

p,q and conversely.

THEOREM 11 C. The transformations T and T-1 in two subscripts are unique in the same sense that the transformations in one subscript were.

The transformation in one subscript can be obtained from that in two by making the second subscript always zero. The theory of the transformation holds of course with any number of subscripts. It is the case of two subscripts we shall use.

IV. THE NUMBERS fij.

12. Definition of the fi. In this part we let the elements a, b, c, ..., of Part III be graphs GI, G,, Gs, ***. If GI and G2 are any two graphs, let Ga + G2 be their sum.




THEOREM 12A. Given any two graphs GI and G2,

(12.1) filj(GI + Gj) --- mpq (GI) mi-p,j-q (G2). p,q

For let HI be any subgraph of GI of rank p, nullity q, and let H2 be any subgraph of G2 of rank i -p, nullity j - q. Then HI + H2 is a subgraph of G?+G2 of rank p+(i -p) = i, nullity q+(j-q) =-j. Con- versely, any subgraph of GI + G2 of rank i, nullity j is made up of a subgraph H1 of GI and a suibgraph H, of G, (either of which may be the null graph), whose ranks and nullities add up to i and j respectively, From this the theorem follows.

Definition. Given any graph a, we define the numbers fij (), (i,) t (0, O) by the transformation T-1:

(12.2) f.j = z ( 14(m) Ic k

As moo = I always, Theorem 1 B, Part III gives the THEOREM 12 B. Given any two graphs GI and G2,

(12.3) f(I (GI + Gh) fij (GI) +fij (GO)

THEOREM 12 C. fij (G) for any graph G is given by a unique polynomial in the numbers N1, Nk, . -, containing no constant term.

This follows immediately from Theorem 7 A, Part II, as fij is a polynomial in the mij containing no constant term.

13. Generalized graphs. With any graph, or real graph, is associated a finite set of integers, positive or zero, N1, N2, -* *, N . Consider now any finite set of integers, positive, negative or zero, N1, N2, . .., Nq, each Nj corresponding to a definite type Tj of non-separable graph. This collection of numbers we call a generalized graph G; we shall say that G contains Ni subgraphs of type Ti, i = 1, 2, ..., q. We may of course add to or remove from a generalized graph any number of numbers which are zero without altering it. We may consider a real graph as a special case of a generalized graph.

Consider now two generalized graphs G and G', whose numbers are N an nd Ni. We let GC+ G', the sum of these graphs, be the set of numbers (13.1) Ni + N;, N2?+ N2, * .., Nq + Nq,

where q is large enough so that all the numbers of both G and a' are included. This holds of course if G and G' are real graphs. Similarly for the difference of two generalized graphs, or any other linear combination.



708 H. WHITNEY.

Various generalized graphs may be expressible in terms of real graphs. This is, in fact, true for all generalized graphs, as the following theorems show.

THEOREM 13A. Given any type of non-separable graph Ti, thegeneralized graph G containing one subgraph of this type, and none of any other type, (Ni = 1, Nj = 0, j t i), can be expressed as a linear combination of real graphs.

Let T1, T2, ..., Tk be all types other than Tj of non-separable graphs which are subgraphs of a graph of type Ti. We arrange them in order, choosing first those with a single arc, next those with two arcs, etc. Types with the same number of arcs we arrange in any order. Then no graph a' of type Tp contains any graph G" of type Tq as a subgraph, q>p.

Let G1 be a graph of type To. Suppose G6 contains c2 subgraphs of type Tk. We then subtract from GI c2 graphs of type Tk, forming the generalized graph C2. In doing this, we have not altered the number of subgraphs of type Ti. C2 contains therefore one subgraph of type Ti, and zero subgraphs of type Tk. Suppose G2 contains cs subgraphs of type Tk_1. Subtract then cs graphs of type Tk_-, etc. This process gives us the required graph.

Example. Let G32 be the graph whose arcs are ab, ac, bc, bd, cd, and say G32 is of type T32. G32 contains a quadrilateral. Let G31 be a quadrilateral; then G32 -G3 is a generalized graph containing no quadrilateral. This graph contains two triangles. Let G21 be a triangle; then C32- 031-2 G21 contains no triangles. As G32 contains five arcs, G31 contains four, and G21 contains three, G32 - G31 -2 U21 contains 5-4 -2 * 3 -5 arcs. Let G10 be an arc. Then

a = G32-G31-2G21+5Glo

is a generalized graph containing one subgraph of type T32, but none of any other type.

THEOREM 13 B. Any generalized graph, whose numbers are N1, N2, * Nq, may be expressed as a linear combination of real graphs.

Let Gi be a generalized graph containing one subgraph of type T2 and none of any other type. We need merely express each Gi in terms of real graphs, and put (13.2) G = NiGi+N2G2 + ..*+NqGq.

14. The numbersfij for generalized graphs. Given the generalized graph G = (N1, N2, .., Nq), we definefij (C) by the polynomial of Theorem 12C. mnh(G) may of course be defined by the polynomial of Theorem 7A. It is




to be noted that, for generalized graphs, there will be in general an in- finite number of non-zero fij and mij.

THEOREM 14A. If G is a generalized graph, made up of the real graphs GC1, G2 , Gj:

(14.1) G= (+ ?*+ Gk- Gk+-1.-..-C1, then (14.2) fij (G) = +(GI) + +fij (Gk) -fij (Gk+1) - -fv (Ga).

By Theorem 12B, this is true if there are no minus signs present, for then a is a real graph. Assume it is true if there are fewer than I- k minus signs present; we shall prove it for the case of I -k minus signs.

Put G' = G+Gi? G +Ck-Gk+l *i Gi-i.

As 0' contains but - k-1 minus signs,

(a) fij (G') = fj (GI) + ** +fij (Gk) -fij (Gk+)- -*fij (GI-,).

Let us add to 0' s graphs congruent with GI:

a(s)= atG'+G1?.?Gt = G'+sGi.

As only - k-1 minus signs are used in expressing this graph in terms of real graphs, we find for s > 0

fij (G'+ s =) fij a) + sfij (=i).

Let N1', N2!, ..., be the numbers for 0', and let N1", N2"', ..., be those for Gz. Then the numbers for G'+ s GI are

Ni'+ s Ni", N2'+s N2", * ,

whether s is positive or negative. Considering N1', N2', ***, N1", N2", ***, as constants, fij(GT'+sG) is a polynomial in s alone:

fij(G'+sGI) = P(s), with the property that

P(s) =fei (a') + s ij (G) =A + B s

if s is any positive integer or zero. Therefore

P(s) = A+Bs.



710 H. WHITNEY.

Now fA(GI'+ s GI) P(s) whether s is positive or negative. Hence, putting s - 1, we have

f,3(G) =A -B =ft(G')-

which, with eq. (a), gives the desired result. THEOREM 14B. Let G and 0' be any two generalized graphs.- Then

A (G + G0) = f (G) +fi (Ga).

Express 0 and G' in terms of real graphs as in Theorem 14A. 0+0' in terms of real graphs is the sum of these expressions. fij () is as given in Theorem 14A, and similarly for fij (0'), fJ (0 + 0'). Comparing these three expressions gives the above equation.

The theorem obviously holds for any linear combination of generalized graphs.

THEOREM 14C. fij (G), a polynomial in the numbers N1, * , Nq, is linear and homogeneous.

Let us put fij (() -P (Ni N2, v Nq).

Let Sk be a generalized graph containing one subgraph of type Tk and none of any other types, k = 1, 2, *-, q. Put

(14.4) ak = fl (Gk) = P(O, * , 1, *..., O),

the 1 appearing in the k'th place. Define the graph 0 by the equation

a N I alN +X2G2+ ***+ Nq Gq,

the numbers N1, ..., Nq being any set of integers, positive, negative or zero. Then

fi(G) N1 fi (a) + N2 fi (G2 + *+ Nq fij (Gq)

by the last theorem, and thus

P(Nj, N2, *, Nq) =a N1j+a2 N2 -+-- +aq Nq.

As the polynomials on the two sides of this equation are equal for all integral values of the variables N1, ..., Nq, they are identically equal:

(14.5) fij (G) _ a, Ni + as N2+. * *+aqNq,

and the theorem is proved. We can, if we wish, express the polynomial fij (G) in the form

(14.6) fV (G) =a N +at +




writing down a term corresponding to every type of non-separable graph. All but a finite number of the coefficients will be zero. We can find any coefficient by the

THEOREM 14D. The coefficient ak of Nk in fij(G) is given by eq. (14.4). This is true if G in terms of real graphs contains the term Gk, as we

have just seen. If it does not, we must show that ak = 0. Take the expression for G and add the term 0. Gk. The proof of Theorem 14 C still holds, and the right hand side of eq. (14.5) now has the added term akNk. This can only be true if ak = 0.

15. Relation between the mij and the Jo;. THEOREM 15 A. Consider the expression for mi (G) for any graph G as

a polynomial in the numbers N1, A, Xq. fij (G) is exactly the linear terms of this polynomial.

By definition,

(15.1) fii(G) = mij(G)+?L (1)k-1 Ri[m(G)]. k_2 k

If we express the right hand side in terms of the numbers N1, N2, , we obtain the polynomial for f0j(U), as this expression in unique, by Theorem 12C. Hence, by Theorem 14C, all terms vanish but the linear terms. The sum on the right hand side is a polynomial in mpq (G) containing no constant or linear terms, and each mpq (G) is a polynomial in N1, N2, * * *, containing no constant terms. Hence the sum contains no linear terms in N1, N2, *.., and thus the only linear terms are those in mij(6), which proves the theorem.

This fact is important for the following reason. If we know the linear terms fpq of mpq for all numbers p ? i, q < j,

then we can calcuate mij, using the transformation T. We shall make use of this fact in the next part. Example. Let us find f3o for graphs containing no 1- or 2-circuits by

eq. (14.5). Let G be a graph containing a single arc, let G' be a triangle, and let G" be a generalized graph containing one triangle and no other subgraphs. Then

G" G'- 3 G.

As a single arc and a triangle are the only types of graphs that can contribute to f3o (cf. Part II),

fso =fso(G)E+f30(G`)N21

- f30 (m E + [f 30(') -3 f30 (G)] 1\J21,

if N21 is the number of triangles in G, by eq. (14.5). Now



712 H. WHITNEY.

310 =3 in80 - in0 i20+ A- mo0, and

mlO(G) = 1, m20(G) = 0, m3o(G) = 0 m 0(G') 3, m20 (G') 3, m80 (G') 0.

Hence

fso(G) - , fo(G') - 9+9 = 0, and thus

fso E N21. 3

Thence, as flo = E and f2 - E

M0ayo 40o+4040+ 6 fi? (3) 2

Let us find a relation between the fij analogous to eq. (1.5), Part I. To do this, we replace each term in this equation by its expression in the numbers N1, N2, . * * (the right hand side is already in that form). By Theorem 5A, Part II, this equation, being true for any graph, is an identity. Therefore in particular, the linear terms on the left hand side equal the linear terms on the right hand side; that is,

(15.2) f40+fi_1,1+ *i+fi () - E.

16. Independence of the m0 and fi,. We give here some theorems on the mij and A} similar to Theorems 5 A and 5B, Part II, on the As.

THEOREM 16 A. If we except the numbers mioo i + 1, there is no polynomial in the remaining mij which equals 0 for all graphs and yet is not identically 0.

If we consider only graphs with no 1- or 2-circuits, we consider only mj which are not 0 for such graphs. (Thus for instance, we do not consider mil.)

Suppose there were such a polynomial. Consider then such a polynomial P in the least number of variables. Of all the numbers Mki appearing, let mij be that variable, chosen from the set for which the first subscript is as great as possible, for which the second subscript is the greatest. Let us write the polynomial in the form

P= Qoma + Q ma-? +** + Qa_1 mij+ Qa,

where the Q's are polynomials in the remaining mki, and QO is not zero for all graphs.




Express now each of the mkz in terms of the numbers N1, N2, * . By Lemma F of ? 8, Part II, there is a non-separable graph of rank i, nullity j, which contributes to mij with the coefficient 1; it contributes to none of the other mkI. Let us say there are Nij subgraphs of this type in a given graph G. Then if Np is the set of numbers excepting Niji

my = NV+R(Np). Let us put

Qs(mkl) Q(Np)

where Q, does not contain Nij. Now

P Qo [Nij + R]a + Q' [Nij + -Rla-1 + ... + Qa

Qo N ?J + (a Qo' R + Qi) Noj + * .. 2

and Nij appears only where shown. By Theorem 5A, Part II, this polynomial, being 0 for all graphs, must vanish identically. Hence the coefficients of the powers of Nij, in particular, Q', must vanish identically. Hence Qo = Qo equals zero for all graphs. We have here a contradiction, and the theorem is proved.

THEOREM 16B. Theorem 16A holds if the words "for all graphs" are replaced by the words "for all non-separable graphs".

We merely use Theorem 5B instead of Theorem 5A. THEOREM 16C. Theorems 16A and 16B hold if "mij" is replaced by 'i4". We replace "mzj" by "fj" throughout the proofs. We of course cannot include mio or fio, i 4 1, in these theorems, as

they are determined from the other mid or fkl by eq. (1.5) of Part I or eq. (15.2) of this part.

17. On inequalities. The theorems of the last article tell us nothing about the existence or non-existence of inequalities on the mij or f,. There are many such inequalities, as we now show.

A fundamental inequality is the obvious one:

(17.1) mni ? 0.

Another set of inequalities is given by eq. (2.1), Part I. Again, the number of ways of coloring any graph is > 0:

(17.2) ,j v-ij( )ifjMj> ? 0 i j

if I is an integer ? 0. Inequalities on the mi may be deduced from the interpretation of the

number (- )tmi in terms of the broken circuits of the graph. We give one example:

(17.3) mi-il - E7



714 H. WHITNEY.

To prove this, take each of the i mi I subgraphs of G of i arcs containing no broken circuit, and drop out any arc. We form thus subgraphs of - 1 arcs containing no broken circuit in i I mi I ways. Consider any such

subgraph. It was formed by dropping out one of the E- (i-- 1) arcs in the rest of G, and was thus formed at most E- i + 1 times. From this the inequality follows.

Further, some polynomials in the fij (or mij) equal a number Nkl, which is of course ? 0. For instance, for graphs containing no 1- or 2-circuits,

fsl -3N21?+N81, Alf N21. Therefore (17.4) N81 =fsl+3f2l > 0.

Of course any inequality on the fV gives rise to an inequality on the mij, and conversely.

A fundamental problem, and one of extreme difficulty, is that of dis- covering all the inequalities.

V. CALCULATION OF THE fij AND Mij.

18. Graphs containing no i- or 2-circuits. In such graphs, moi ml= 0, or, fol = fi = 0. If there is but a single type of non-separable graph of rank i, nullity j, we shall call this type TVj; if there are several, we shall call them 'Tij, 2Tij, *. * * qTij. Define Nj, or Nij*., qlj correspondingly. If there is more than one Tij, put Ni - kNij. Thus Nij is the number of non-separable subgraphs of

k

rank i, nullity j. The non-separable graphs of ranks one to four are given in the figure.

I A DKI \2?ZK T10 T21 T31 T32 T33 T41 1T42 42

I T43 2 T43 3 T43 I T44 2T44 T45 -T46 Fig. 2.

We shall calculate the fj in terms of the kNpq in the following order: fo, f20, f21, fAym f31, **, f.. , *g.. To find an fij, having found the



THE COLORING OF GRAPHS. 715 previous fkl, we first write down mij in terms of the fkz by eq. (11.2), Part III. We know all the fkl occurring in this expression except fu. Now, by eq. (14.5), Part IV,

fy = ail Nil + as, Ni, + ***+ asp Ni

where the corresponding types Til, * i, T are restricted as seen in ? 8, Part II. Let Gil, *., Git be non-separable graphs of these types, arranged in such an order that no one contains any one following as a subgraph. Consider first Gil, and let its numbers be N1 (Gil), N2 (Gil), **, Nq (Gil). Substitute these values in the expression for mix. mij (Gil) will be 0 if Gi1 is not of rank i, nullityj, and N2,, * *, Ni are all 0 for Gsh. Hence in the resulting expression ail is the only unknown, and we can solve for it. We can now find ai,, using Gi,, etc.

Several facts will aid us. If Gi, is of rank i, nullity j, then aik = 1. If j>0, then alo, the coefficient of E = N10, is 0 (see Theorem 8A, Part II). fjo, i> 1, is always most simply determined by eq. (15.2), Part IV. We turn now to the calculations.

[1,0] m10o=E, A0= E. 1 [2,0] f2o+fl +fo2 f2O E

2,1] m2= N21, f2l N21i

[3 01 E f -1-1 [3,0] fao?f21 = pE, 2.fso = 2E-N1.

[3,1] Mt = flof2l +fsl EN21 +(cN21 +N81).

For a graph G21 of type T21,

m81 = 0, E 3, N21 1, N81 0. 0=3.1?c1, c 3. f8s 3 N21 + N81.

[3,2], [3,31 f82 = N83, f -s N88.

[4,0] f40+fsl = -LE, ..f4o =- E+3Ns-N 4 4

m41 (-2 ii0f2l +flf81 +f2of21) +f41

[4,1] l(2 XN21-3 - EN2?+EN N)

+(cN21+dN81+ e N82+N41). 47.



716 H. WHITNEY.

021: E = 3, N21 1.

0 .3211 313.1+c, C 6. 2 2

Gs,: E = 4, 81 - 1. .'. 0 = 4+d, d 4.

G32: E = 5, N21 = 2, N31 1, 82 1.

0 = 25-35+5+12-4+e, e 3.

.f41 = 6N21-4 N31 -3 N82 + N41.

Similarly,

f42 = - N21-6 N82 -6 N8+N42,

=s 6N8s + N4s, f4j = N4j, j = 4, 5, 6;

f5 5 E- 6N21?+4N8+2N82-N41,

=A 9N21+ 1ON831+ 17 N82?+11 N8-5N41-31N42

-32'N42 + 51,

(18.1) f52 3N21+19N82 + 36N88 -7 1N42- 62N42-5'N48 - 6 2N4s - 7 8N48 + N52,

58- 2N82 + 9+N88 -9 'N48-7 2N48-108N4s-81N44 - 8 2N44 +N5,

f54 -z--4N8 8 o 8 1N44- 9 N44- 9 N45 +N54,

A 55 -9 N456 + N,55,

f56- 1ON46+N56, f5j N6j, j 7,8,9,10.

From these formulas we can write down the mij. Thence we find the

mi, the coefficients of M(2). The first few are

MO 1, ml -E, m2 = 2- N21

(18.2) = -(3) + E-2'N21?N31-N32?N33.

We can now find the ai from eq. (4.4), Part I. The first six are

ao0 1, a1 = 0, a2 N21, as Ns1Vs2 N,

a4 3 Nss + (- 1)j N4j, (18.3)

=f? N21 (NA1 N82 + Nss) + 6 Nss + 2N42 3 2N48 -N48 10

+ 4 1N44 + 3 'N44- 4 N46 + 4N46- (- 1)j N5j.




From these formulas and eq. (4.4) we can calculate the expressions for m4 and m5, which are rather lengthy.

19. Regular maps. It is well known that in studying the four color map problem, it is sufficient to restrict oneself to certain special maps; in particular, maps which we shall call regular, obeying the following conditions:'0 (1) All the vertices are triple. (2) There are no rings of one, two, three or four regions; that is, no set

of four or fewer regions, together with all boundary lines separating them, form a multiply connected region.

(3) There are no rings of five regions except about a single region. The numbers mij have been defined in terms of the dual graph G of

the map. G', the graph consisting of the boundary lines and vertices of the map, is the dual of G in the sense defined in N. We can state the above conditions in terms of G and 0' as follows: (1') Each vertex of G' is on exactly three arcs. (2') G is quintuply connected." It follows that G contains no 2-circuit.

For otherwise, dropping out the two corresponding arcs of G' would separate it into the pieces H1' and H2, each of which is seen to be of nullity >0. But from this it could be shown that dropping out the two vertices of the 2-circuit in G disconnects G." Consequently G and G' determine each other uniquely.-1

(3') If dropping out five vertices from G disconnects G, one of the resulting pieces contains just one vertex.

The numbers N1, N2, ** *, involved in m1, ***, m5, can be expressed in terms of the following quantities for all regular maps:

V = no. of regions in the map (no. of vertices in the dual graph).

E 3(V- 2) = no. of boundary lines (no. of arcs).

T = 2(V- 2) = no. of vertices (= N21).

Q5 no. of 5-sided regions (no. of vertices on five arcs).

R6 = no. of rings of six regions, no two having a common boundary unless they are adjacent in the ring (no. of 6-circuits in which only vertices adjacent in the circuit are joined by arcs in the graph).

We derive the following formulas.

10 See G. D. Birkhoff, The reducibility of maps, American Journal of Mathematics, 35 (1912), pp. 115-128.

I" See C. G., ? 2 (p. 158). 12 Loc. cit., Theorem 9. 13 Loc. cit., Theorems 10 and 11.



718 H. WHITNEY.

1 2 fo=E, f20 - 2E, Al E

-so E, f -- E, f32= E, 3'

(19.1) f4o 3

El f41 =-E+Q5, f42 =-2 1 E, f48 = 2E,

1 2 f50-D E-Q5, f5l 3-- E+R6, f52 = 3E+15Q5,

As -6E+10Q5, A4 = 4 2 E+5Q5, fsi Q5a

=O 1y ml -E, M2 i n E) T

(1 9.2 +(E -2)Ty

(19.2) (E) (E-2) T+ (T)Q

m6w E- + (E32 T- (E- 4) (T +(E -8) Q5+ R6; fl13 =

3

a0 = 1, a1 0, c2 -Ty as = 0, (19.3) a4 a a5 = 4Q5-R6.

The regular map of the fewest number of regions is the dodecahedron, containing twelve regions, each touching five others. For this map

V 12, E 30, T= 20, Q5 12, Re - 40.

We can thus calculate the first six coefficients. The whole polynomial, which was calculated by direct means, is

M(A) = A(A 1)(A-2)(A-3)[A8-24A7+260A6-167OA + 6999 i- 19698 As + 36408 2- 40240 A + 20170].

NOTE ADDED IN PROOF. R. M. Forster has calculated the mu for a large number of graphs, using the following formula. If G contains the arc a b, G' = G - a b, and G" is formed from G' by letting a and b coalesce, then

(19.4) my (G) = mv (G') + mi-ij (G"), as is easily proved.

PRINCETON UNIVERSITY.



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