Tulgeity of Line Graphs

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Joum al of Mathematics Research ISSN: 1916-9795Vol. 2, No . 2, May 2010 E-ISSN: 1916-9809

Tulgeity of Line GraphsAkbar Ali.M.M (Corresponding author)

Department of M athematicsSri Shakthi Institute of E ngineering and TechnologyCoimbatore 641 062, Tamil Nadu, India

Tel: 91-999-427-9835 E-mail: um-akbar@ yahoo.co.inS.Panayappan

Department of Mathematics, Govemment Arts CollegeCoimbatore 641 018 , Tamil Nadu, India

Tel: 91-944-310-9345 E-mail: panayappan@g mail.com

AbstractTulgeity T { G ) is the maximum number of disjoint, point induced, non acyclic subgraphs contained in G. The formula forthe tulgeity of the line graph of com plete graph and complete bigraph are derived. A lso we present an upperbound for thetulgeity of line graph of any graph and w e classify the graph for which the upperbound becomes the formula.Keywords: Tulgeity, Line graph1. IntroductionPoint partition num ber (Gray C hartrand, 1971) of a graph G is the minimum number of subsets into which the point-setof G can be partitioned so that the subgraph induced by each subset has property P. Dual to this concept of point partitionnumber of graph is the maximum number of subsets into which the point-set of G can be partitioned such that the subgraphinduced by each subset does not have the property P. Define the property P such that a graph G has the property P ifG contains no subgraph which is homeomorphic from the complete graph ^^3. Now the point partition number and dualpoint partition number for the property P is refered to as point arboricity and tulgeity of G respectively. Equivalently thetulegity is the maximum number of vertex disjoint subgraphs contained in G so that each subgraph is not acyclic. Thisnumber is called the tulgeity of G denoted by T { G ) . Also, T ( G ) can be defined as the maximum number of vertex d isjointcycles in G. The formula for tulgeity of a complete bipartite graph is given in (Gray Ch artrand., 1968). The problems ofNordhaus-Gaddum type for the dual point partition numb er are investigated in (A nton, 1990).All graphs considered in this paper are finite and contains no loops and no m ultiple edges. Denote by [x] the greatestinteger less than or equal to x, by |51 the cardinality of the set S, by E { G ) the edge set of G and by K the complete graphon n vertices. The other notations and terminology used in this paper can be found in (Frank Harary, 1969).Line graph of a graph G is defined with the vertex set E { G ) , in which two vertices are adjacent iff the corresponding edgesare adjacent in G. Since T { G ) < , it is obvious that T { L { G ) ) - . However for complete graph Kp, T(Kp) = rr .

L ^ J L.JJ L.^J2. Main TheoremsWe now present a formula to find tulgeity of line graph of a complete graph.Theorem 2.1. The tulgeity of line graph of complete graph Kn, T(L(')) =ProofLet V(A'n) = {vi,V2, . . . , v )an dl ete y be thee dg e joining the vertices V, an d vy. i .e. ,(ir) = {e,; : 1


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ISSN: 1916-9795 Jou ma l of Math ematics ResearchE-ISSN: 1916-9809 Vol. 2, No . 2, Ma y 201 0

ViK\) = {eij : 2 < j < n]

ViK) = {ei4, e24, e34} U {e4j : 5 < j < n]

In genera l ViK) = ( e , , : I < i < I-l]U {eij : 1+I < j < n]C a s e i):n= limod 3)(n - 1) is a multiple of 3 and henc e the order of each cliqu e of LiK) is a mutiple of 3. In this case the disjoint cycles ofLiKn) are counted as follows. C onsider the following set of cycles of ATJ, A' j , . . . , K'n.Let Ci = [e\2e\-ieu,eiie\(,e\i,...,e^n-2)e\(n-\)e\n] and let V(Ci) be the set of vertices belonging to the cycles of C\ .Clearly the cycles of Ci are disjoint and y(Ci) = ViK\). Also |Ci | = '^-^.Let C2 = {e25e26e2i,e2%e2ge2\o,...,e2(n-2)e2(n-i)e2n] and let VC2) be the set of vertices belonging to the cycles of C2.Clearly the cycles of C2 are disjoint and VC2) = ViK'^) - {^23, ^24, e^], where en e ViK[). A lso IC2I = '^^-.Le t C3 = {e35e36e37, e38e3 9e3io,...- e3(_2)e3(-i)e3) a nd let VC3) be the set of vertices belonging to the cycles of C3 . T hecycles of C3 are disjoint and VC3) = ViK^) - {^34,623,^14}, where ^23 e ViK'^) an d e^ ViK[). A l s o IC3I = ^ ^ .Now the vertices 23. 24 and 634 form a 3-cycle in LiKn). Continuing this process we get IC4I = '^-^, IC5I = ^^,

J .3ICfil = ^ , , |C_ 5| = 1, |C_4 | = 1, |C-3 l = 1, |C-2l = O, |C _ i| = o, \Cn\ = 0.Le t C+i be the set of cycles formed by the ve rtices of ViK) - ViV'.), (2 < / < n) .n- li.e., CH.i = < e,(,+i)e,(,+2)e(i.n){,+2) : / = 3fe - 1, 1 -^ -.6 6C ase ( ii ) : n = l im od 3)in - 2) is a multiple of 3. In this case the disjoint cycles of LiKn) are counted as follows. Consider the following setof cycles of K\,K'.^,...,K'n. Let C, be the set of cycles of the clique K' . (1 < < n) and let V(C,) be the set of verticesbelonging to the cycles of C,.Let C | = {ei3ei4ei5,ei6ei7ei8,. . . ,ei(_2)ei(_i)ei | , where V(Ci) = ViK[)- \Let C2 = 1^23624^25. C26e27e28>.... e2(n-2)e2(n-i)e2nl. whereLet C3 = {36E37^38. ^39^310^311..... 3(n-2)63(n-l)C3n}, whereLet C4 = {en(,en-]e4%,e,ge4\Qe4\\,...,e4(n-2)^4(,n-i)S4n], w h e r e VC4) = ViK',) {45}.Clearly the cycles of Ci, C2, C3 and C4 are disjoint an d the v ertices 634,635 and 45 form a 3 -cycle . Cont inuing this processwe g e t I C I = ^ , IC2I = ^ , IC3 I = ^ , IC4I = ^ , IC5I = ^ , , |C_5| = 1, |C_4| = 1, |C_3| = I . C-2.Cn-\ and Cn are empty. Let Cn+\ be the set of cycles formed by the vertices of ViK'.) - V(C ' ) . (3 < ( < n) .i.e.,C .n = |e,{,+i)e,{,+2)e(,+ixi+2) : ( = 3:, 1 :< : < - \.Clearly C+i contains disjoint cycles and |C+i| - -. Henc e minim um number of cycles of LiKn) is |C,| =3 ;


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Joumal of Mathematics Research ISSN: 1916-9795Vol. 2, No . 2, May 2010 E-ISSN: 1916-9809Case (iii) : n s Oimod 3)n is a multiple of 3. In this case the disjoint cycles of LiKn) aie counted as follows.Let C, be the set of cycles of the cliqueK il [ ^ ]Now by the very definitions, LiKn) has vertices and hence TiLiKn)) < = ,Hence rmKn))=[^].Each vertex v, of a graph G w ith degree de g v, induce a clique of order de g v, in LiG). i.e.. At each vertex v, of G, wecount -fi^ vertex disjoint cycles in LiG). The above argument yields the following result which gives an upperboundfor the tulgeity of line graph of any graph.Theorem 2.2.

The following result characterize the graph for which the above said inequality becomes equal.Theorem 2.3. If G isa tree and for each pair of vertices (v;, v^) with de g v,, deg vj > 2, if there exist a vertex v of degree2 on Fivi, vj) then TLG)) =Froof Let T be a graph satisfying the given condition. Each vertex v, of T of degree > 2 induce a clique of order de g v,- inLiG) and since for each pair of vertices (v,, vj) of T with degree > 2, there exists a vertex v of degree 2 on / '(v,, v;), there[ deg Vi 1The tulgeity of complete n-partite graphs have been studied by Gary C hartrand et.al in (1968). Here w e derive a formulafor the tulgeity of line graph of complete bigraph.Theorem 2.4. The tulgeity of line graph of complete bigraph grap h Km,n, riLiK,,)) = "" Froof Without loss of generali ty assume that m < n. Let ViKm,n) = {vi, V2, ., v) U ( i, M 2 , . , , ,) and EiK,n) - [eij =ViVj : I < i < m,\ < i < n\, |(/fm,)| = mn. By the definition of line graphs, ViLiK^^n)) = {eij : I < i < m,l < j < n]in which ei j and ek i are adjacent if either i = k or j = I. Hence \ViLiKm.n)\ = mn and hence TLiKm,n)) < ^ .Let V, = {ej : I < j < n],il < i < m). Clearly


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ISSN: 1916-9795 Joumal of Mathematics ResearchE-ISSN: 1916-9809 Vol. 2, No. 2, May 2010

If n = l{mod 3) then V, - {,), (1 < / < m) has n - 1 vertices for each i and there exist vertex disjoint cycles inCase (ii):

n = l{n- {eij] for each i. Also, since (e^, e2,.-, e^) is a complete subgraph there exist vertex disjoint cycles. Hence

/ 1 \ f - - l r ' - i L . J J

Case (iii):If n s 2{mod 3) and m = l{mod 3) then V, - ({e,) U (e,(_i))), (1 < :< m) has n - 2 vertices for each / and there existvertex disjoint cycles in V,- - ((e,} U {e,(_i)}) for each /. Also since m s l{mod 3), hasvertex disjoint cycles and (ci,e2, ...,c) has ^ ^ ^ vertex disjoint cycles. Hence T{L{Kn,,n)) > ^^" ~ ^^ + ^m - 1 mn-2 Tmni

. n - 2m - 1

3 ~ 3Case (iv):If n s 2{mod 3) and m = 2{mod 3) then V,- - ({e,) U {e,(_i)} U {e,(_2)}), (1 < ( < m) has (n - 3) vertices for each i and

. n - 3there exist -y- vertex disjomt cycles in V,- - ((e,} U {e,(_i)) U {e,(_2)}), (1 < < m) for each i. Also m = 2(mod 3),/ \ u ^ ~ 2 ' /?z 2\ei(_i) , e2(-i),... e(;-2)(-i)) has J - vertex disjoint cycles and (ei , e2,.., e(m-2)n) has vertex disjoint cycles. Atlast the vertices e(-i)(-i), e(_i), (_,), e, induces a 4-cycle. Hence T ( L ( / S : , ) ) > ^^" ~ ^^ + 2 ^ ^ ^ H- 1 = ^ " ~ ^ =

^ ] . Hence in all the cases T ( L ( ^ , ) ) > [ ^ ] . Hence T(Z.(i:,)) = [ | ^ ] .ReferencesAnton Kundrik. (1990). Dual point partition number of complementary graphs. Mathematica Slovaca, 40(4), 367-374.Frank Harary. (1969). Graph Theory. New Delhi: Narosa Publishing home.Gray Chartrand., Dennis Geller., & Stephen Hedetniemi. (1971). Graphs with forbidden subgraphs. Joumal of Combina-torial Theory, 10, 12-41.Gray Chartrand., Hudson V. Kronk., & Curtiss E. Wall. (1968). The point arboricity of a graph. Israel Joumal ofMathematics 6(2), 168-175.


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Tulgeity of Line Graphs