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n-connected graphs for fixed n [3,5], and both minimum and maximum critical n-edge connected graphs for fixed n. In this paper we investigate the maximum critical n-edge connected graphs. In [ 11 we investigated the minimum critical n -edge connected graphs.

Let n be a fixed integer, n 2 2. Let 9’ be the set of all critical n-edge connected graphs, and let 3 be the subset of Y in which every graph is n-nbd- regular. Krol and Veldman characterized the set of all maximum n-nbd-regular critical n-connected graphs, denoted d. As we will see in the next section, the set of maximum members of d is contained in the set of maximum members of 9. We will not only determine the maximum members of Y, but we will show that the number of edges is the same, whether one is talking about maximum members of 3 or d.

We use [XI to denote the greatest integer less than or equal to x .

2. CHARACTERIZATION OF THE MAXIMUM MEMBERS OF 9

Fix n. To characterize the maximum members of 3 we need a few easy lemmas.

Lemma 1. d C 3.

Proof. Let G E d. G is n-nbd-regular; therefore for any vertex u, G - u has a vertex of degree n - 1. Therefore, h(G - u) = n - 1. Hence G is criti- cal n-edge connected and G E 3. I

(The converse is not true, as shown in Figure 1.)

G:

u4 FIGURE 1

CRITICAL n-EDGE CONNECTED GRAPHS 561

Lemma 2. If n = 2, then d = 9.

Proof. G E 9. Therefore, X(G) = 2, and X(G - u) = 1 for all u. Since K ( G ) I X(G) = 2, K(G) = 1 or 2. Since one vertex cannot disconnect G and X(G - u ) = 1, K ( G ) = 2, and K(G - u ) = 1. Therefore, G E d.

We want to construct n-edge connected graphs with m vertices, m 2 n + 1. Y C d . I

For n + I I m 5 2n, construct the graph H,,, as follows:

V(H,) = {uI,u2, . . . ,urn}, where {u3, u4, . . . ,urn} induce a complete graph,

and for m = n + 3, n 2 3, the graph U,,, as follows:

Figure 2 illustrates H, and U, for n = 4. It is easy to see that both H, and Urn are in 9, and that each has (2n - 1) +

("i2) edges. Denote the set of critical n-edge connected graphs in 9 with m vertices by 9,,,, and the set of maximum critical n-edge connected graphs in 9,,, by M(T,,,). Denote the number of edges of a graph G by v (G) .

H, : u, :

" 5

FIGURE 2

U " S

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564 JOURNAL OF GRAPH THEORY

Subcase i . If (k, - k ) (k, + k - 2n) = 0, then either k, = k or k,,, + k = 2n. Since k, 2 k 2 n, in either case k = k, as desired.

Subcase i i . If (k, - k ) (k, + k - 2n) = 1 , then both factors must equal 1 since both are integer valued. Therefore, k, = k + 1 = n + 1 and k = n. k, = [((n - l)m)/n] or [((n - l )m)/n] - 1 by definition. In either case we have n + 1 I ((n - l)m)/n I (n + 2). Since 2n + 1 I m I (n(n + 2)) / (n - l ) , (2n + 1 ) (n - 1) I n(n + 2). Therefore, 2 5 n 5 4, and n = 2 or 3. n # 2; hence n = 3. Thus, 7 I m I (15/2), and m = 7, with k = k, - 1, as desired.

Case2. k = n - 1 . Sincem 2 2n + 1 andn # 2, k, 2 n + 1. Therefore,

With equality we have [(f,(k,))/2] = ( f , (k)) /2 . Therefore, as in Case 1, (k, - k ) (k , + k - 2n) = 0 or 1. If both terms are equal to 1 , then k, = k + 1 = n and k, + k = 2n - 1 # 2n + 1, a contradiction. If k, + k - 2n = 0 , then k, = 2n - k = n + 1 = k + 2 , and we are in Subcase ii of Case 1 with n = 3 and m = 7 and k = k, - 2.

Case 3. k I n - 2. By Lemma 3 , IE(G)I I ( ( fm(n)) /2) - 1 . Since n pro- duces the minimum off,, IE(G)I 5 ( ( fm(n)) /2) - 1 5 [(f,(k,))/2]. If equality exists, then [(f,(n))/2] = ( ( f , ( km) ) /2 ) + 1, a contradiction to n producing the minimum.

Therefore, we have shown that IE(G)I I [(f,(k,))/2] for n # 2 , and for n = 3 and m = 7 equality exists only if k = k, or k, - 1 or k , - 2; otherwise equality exists only if k = k,.

The only case that remains is n = 2. But if n = 2 , then d = 3, and the Krol-Veldman results yield the conclusions of the theorem. I

Now we can construct n-edge connected graphs with m vertices where

Construct the graphs Z, as follows: m 2 2 n + 1.

V(Zm) = ( ~ 1 , ~ 2 7 * * * 9 urn} , K(Z,) = {u,, u2, . . . , uh} and (K(Z,)) is complete;

l 2 p3 u

if m - k, is even;

if m - k, is odd;

(m - km) p2 7

( m - k, - 3) P,,

(R(Zm))

each vertex of K(2,) is adjacent to a vertex of R(Z,) .

CRITICAL n-EDGE CONNECTED GRAPHS 565

Construct G , graphs such that each has all of the properties of the 2, graphs plus the additional property: If u and u are vertices of a component of (R(G,)) isomorphic to a P2, then I(N(u) U Nfu)) n K(G,)I 2 n. Therefore, {G,} G {Z,}.

Lemma 4. If G E { Z , } , n 2 3 and rn 2 2n + 1 , then G E 5, with [(fm(km))/21 edges*

Proof. If G E {G,}, then G E d with [ ( f , (k , ) ) /2] edges by [ 5 ] . There- fore, G E 3,.

If G G {G,}, then there exists vertices u and u of R ( G ) such that I(N(u) U ~ ( u ) ) n K(G)I I n - 1 . Since deg(u) = deg(u) = n, degR(u) = degR(u) = I , IN(u) n K ( G ) ( = / N ( u ) n K(G)I = n - 1 # 1 since n # 2 . Therefore, I(N(u) U N(u) ) fl K(G)I = n - 1 and IK(G)I = k = k, 2 n - 1 + 2 = n + 1. h(G) = n and for all x , h(G - x ) = n - 1 . Every vertex is adjacent to a vertex of degree n ; therefore, G E 3,.

IE(G)I = 1 / 2 C degc(v) + C degc(U) (... VEK 1 + $ ( k m - 1 ) ' I (n - 1) (rn - k,,, - 1 ) + (n - 2 ) 1 - Kedges

(n - 1) (m - k m )

or = 1/2(rn - kn)n + R edees

Note: This is the same size edge set as [ 5 ] showed for graphs in d,. Define 3t = {G,,,) U {H,} and 2 = {Z,,,) U {If,,,} U {U,}. Graphs B,, T,, TS, and TY used in the next theorem are pictured in Figure 3.

Theorem 3. If G E 3, and rn 2 n + 1 , then u(G) = [cf,(k,))/2]. Moreover,

if n # 2, n # 3 i f n = 2 if n = 3 . 2 U {B,,T,,T;,T:'},

Proof. For n + 1 I rn I 2n. We are done by Theorem 1 .

Let rn 2 2n + 1 and consider three cases: n = 2, n = 3 and in = 7, n # 2 , and (n # 3 or in # 7).

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References

[l] M. B. Cozzens and S. S. Y. Wu, On minimum critical n-edge connected

[2] M. B. Cozzens and S. S. Y. Wu, Graphs that are n-edge connected and

[3] R. C. Entringer, Characterization of maximum critically 2-connected

[4] F. Harary, Graph Theory. Addison Wesley, Reading, MA (1969). [5] H. J. Krol and H. J. Veldman, On maximum critically n-connected graphs.

[6] L. Nebesky, On induced subgraphs of a block. J . Graph Theory 1 (1977)

[7] H. J. Veldman, Non-k-critical vertices in graphs. Discrete Math. 44 (1983)

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