CHAPTER 2 DECOMPOSITION OF GRAPHS

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1 CHAPTER DECOMPOSITION OF GRAPHS. INTRODUCTION A graph H s called a Supersubdvson of a graph G f H s obtaned from G by replacng every edge uv of G by a bpartte graph,m (m may vary for each edge by dentfyng u and v wth the two vertces n,m that from one of the two partte sets. We denote the set of all such Supersubdvson graphs by SS(G. Then, we prove the followng results. Each non-trval connected graph G and each Supersubdvson graph H SS(G admts an even and odd α valuaton. Consequently, due to the results of Rosa [9] (Theory of Graphs, Internatonal Symposum, Rome July 966, Gordon and Breach, New York, Dunod, Pars, 967, p and El-Zanat and Vanden Eynden [5] (J. Combn. Desgns 4 (996 5, t follows that complete graphs and complete bpartte graphs can cq+ mq, nq be decomposed nto edge dsjoned copes of H SS(G, for all postve ntegers m, n and c, where q = E(H.

2 3 A functon f s called an Even-Graceful Labelng of a graph G wth q edges. If f s an njecton from the vertces of G to the set { 0,,,...,q} such that when each edge uv s assgned the label f ( u f ( v, the resultng edge labels are dstnct even numbers,.e. ranges from to q. A functon f s called an Odd-Graceful Labelng of a graph G wth q edges. If f s an njecton from the vertces of G to the set { 0,,,...,q } such that when each edge uv s assgned the label f ( u f ( v, the resultng edge labels are dstnct odd numbers,.e. ranges from to q. A graph, whch admts an Even/Odd Graceful numberng, s sad to be Even/Odd graceful graph. An even graceful labelng f s called an Even α - valuaton of G, f there exst an even nteger k, so that for each edge uv ether f ( u k < f ( v or f ( v k < f ( u. It follows that such a k must be the smaller of the two vertex labels that yeld the edge labeled for even graceful and for odd graceful labelng. A graph G admttng an Even or Odd α valuaton s necessarly bpartte. Rosa [9] proved the followng sgnfcant theorem Theorem. If a graph G wth q edges has a α valuaton, then there exst a cyclc decomposton of the edges of the complete graph cq+ nto sub graphs somorphc to G, where c s any postve nteger.

3 4 By decomposton we mean a set of sub graphs, whch parttons the edges. For defnton and detals on cyclc decomposton of cq+ nto sub graphs refer the classc paper by Rosa [9] and EI-Zanath and Vanden Eynden [5] Rosa s theorem provdes motvaton to construct famles of bpartte graphs whch admt an α valuaton. In ths drecton some nterestng results have been obtaned n [7, 8]. For an exhaustve survey on these topcs, we refer to the excellent survey paper [8]. The operaton Supersubdvson of non-trval graph s used to generate famles of bpartte graphs. In the complete bpartte graph,m, the part consstng of two vertces s termed as the -vertces part of,m and the part consstng of m vertces s termed as the m-vertces part of.,m Let G be a graph wth n vertces and t edges. A graph H s sad to be a Supersubdvson of G f H s obtaned from G by replacng every edge e of G by a complete bpartte graph,m, for some m, t (m may vary for each edge e n such a way that the ends of e are merged wth the two vertces of the -vertces part of,m after removng the edge e from G. In secton. of ths chapter, an algorthm (Algorthm. s gven to construct certan Super subdvson of any non trval connected graph. The set of all such Super subdvson graphs of a non trval connected graph G s denoted by SS (G.

4 5 In secton.3, t s proved that there exsts a Super subdvson graph H of G admttng an even graceful labelng. Consequently,by Rosa s theorem t follows that some complete graphs can be cyclcally decomposed nto sub graphs somorphc to such Super subdvson graphs and by the theorem of El-Zanat and Vanden Eynden [5] t follows that some complete bpartte graphs can be decomposed nto sub graphs somorphc to such Super subdvson graphs, where q = E (H, m and n are arbtrary postve ntegers.. CONSTRUCTION OF SUPERSUBDIVISION GRAPHS OF CONNECTED GRAPHS In ths secton an algorthm for the constructon of certan supersubdvson graphs of a non-trval connected graph G s gven. Algorthm.: Basc labelng for the vertces of a connected graph. Let G be a non-trval connected graphs wth n vertces and t edges. Step : Assgn 0 to any vertex v of G. Step : Fnd the least labeled vertex of G, whch has adjacent unassgned vertces. If w, w,., w r are the unassgned adjacent vertces of that vertex, then assgn the numbers s+, s+4, s+6,, s+r, where s+ s the least postve nteger avalable for the assgnment.

5 6 Step 3: Repeat step untl all the vertces are assgned numbers from 0 to (n-. The labeled non-trval connected graph G by usng Algorthm. s denoted by B(G. Fgure. s a basc labeled graph B(G, where G s a connected graph. Remark.: Observe that n B(G, vertces are labeled wth numbers 0,,,(n-. Snce G s connected, each vertex n B(G labeled has at least one adjacent vertex labeled r such that r <, for all except for = 0. Remark.: It s clear from Algorthm. that every non-trval connected graph G has fntely many dfferent B (G s. Fgure. Basc labeled graph B(G

6 7 Algorthm.: Constructon of a Supersubdvson graph of a non-trval connected graph G. Step : Usng Algorthm. label the vertces of G and hence obtan the B(G. Step : Let e, e,..,e r be the edges of B(G. For each edge e = xy of B(G. t defne m = j x y, where j s any postve nteger. Now n B(G, replace each edge e, t by,m by mergng the end vertces of e wth the two vertces of -vertces part of,m after removng the edge e of B (G. Let SS(G denote the set of all Supersubdvson graphs of G constructed by usng Algorthm.. Remark.3: Observe from step of algorthm. that each edge e = xy n B(G s replaced by where m,m = j x y j s any postve nteger. Thus, for each edge e, there are nfntely many choces of replacement n the constructon of the Supersubdvson graph.,m for ts Hence, from each edge of a non-trval connected graph G, an nfnte famly of Super subdvson graphs can be constructed. The famly of Supersubdvson graphs of a connected graph G constructed usng Algorthm. denoted by SS(G.

7 8 An llustraton of a Super subdvson graph of the graph G constructed usng Algorthm. s gven n Fgure.. Fgure. A Supersubdvson Graph of G constructed by Algorthm.

8 9.3 SUPER SUBDIVISION OF A GRAPH In ths secton t s proved that for any non-trval connected graph G each Supersubdvson graph of G n SS(G admts an even graceful labelng. Observaton.. For a non-trval connected graph G, let k denote the cardnalty of the set of all edges of B (G ncdent wth the vertex labeled wth whose other ends have the label less than. Let r,..., r, r k be the vertces of B(G adjacent to wth r < r <... rk <.Then to obtan a Supersubdvson graph H of G. By step of Algorthm., the edges r r,... r, k of B(G are replaced respectvely by j ( r,,, j ( r,.,, j k ( r k Where j, j, j are arbtrary postve ntegers. k Therefore, we have E (H = n- k å ( å (j l ( - r l = l= Theorem..For any non-trval connected graph G, each Supersubdvson graph n SS(G admts an even α valuaton. Proof: let G be a non-trval connected graph and let H be any Supersubdvson graph G n SS(G. Let M be the number of edges of H. By Observaton, we have n- M = k ( (j l ( - r l å å = l=

9 0 For convenence, denote N 0 = M ; and for n- Let N = N0 - ( (j ql ( q - r ql k q å å q= l= A graceful labelng shall frst be gven to the vertces of H and then shown that ths labelng s ndeed an even graceful labelng for H. For the base vertces of H consder ther same basc labels n B(G and for the remanng non-base vertces of H labels shall be gven n the followng way. For each, n- we consder the non-base vertces of the complete bpartte graphs ( r, j,, j ( r,.,, j ( r k k of H whch are obtaned n the constructon of H from B(G by replacng all the edges, r,... rk of B(G such that r < r <... rk <. r, Now for each p, p k labels shall be gven to the non-base vertces of, jp ( rp as shown n Fgure.3 Observe from the labelng that the labels of the non-base vertces n the frst set of, jp ( rp form a monotoncally decreasng sequence and thus the labels of any two non-base vertces dffer by at least one. Hence the labels of all the non-base vertces of the frst set of the (, jp r p are dstnct.

10 Further, observe that the least value of the labels of the non-base vertces of the frst set of (, jp r p p- N -- å (j l ( - r l +( -r p + r p +; l= and the largest value of the labels of the non-base vertces of the second set of (, jp r p p- N -- å (j l ( - r l -( -r p + r p ; l= dffer by -r p + (note that > r p. As n the frst set, the labels of the non-base vertces of the second set also form a monotoncally decreasng sequence, thus the labels of any two non-base vertces of the second set, jp ( rp wll dffer by at least one. Hence the labels of all the non- base vertces n the second set are also dstnct. Smlarly the labels of the non-base vertces of the other remanng j p sets of, ( jp r p are all dstnct, provded that the label