IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 5 Ver. IV (Sep. - Oct. 2015), PP 31-33 www.iosrjournals.org

Decomposition of Line Graph into Paths and Cycles

S.Lakshmi1, K.Kanchana2 1Assistant Professor, Department of mathematics, PSGR Krishnammal college for women, Coimbatore, Tamil nadu, India 2Research Scholar, Department of mathematics, PSGR Krishnammal college for women , Coimbatore, Tamil nadu, India

Abstract: Let P푘+1 denote a path of length 푘 and let C푘 denote a cycle of length 푘. As usual 훫푛 denotes the complete graph on 푛 vertices. In this paper we investigate decompositions of line graph of 훫푛 into p copies of P5 and q copies of C4 for all possible values of p ≥ 0 and q ≥ 0. Keywords: Path, Cycle, Graph Decomposition, Complete graph, Line graph.

I. Introduction Unless stated otherwise all graphs considered here are finite, simple, and undirected. For the standard graph-theoretic terminology the readers are referred to [7].Let denote a path of length and let denote a cycle of length 푘. Let denotes a star on vertices, i.e. = . As usual denotes the complete graph on 푛 vertices. Let denote the complete bipartite graph with 푚 and 푛 vertices in the parts. If G = (V,E) is a simple graph then the line graph of G is the graph L(G) = (E,L), where L = {{푒,푓}|{푒,푓} ⊆ E,|푒 푓| = 1}. For the sake of convenience, we use 푢푣 to denote an edge {푢,푣} and 푢,푣 are called the ends of the edge {푢,푣}. In general, if one allows more than one edge (but a finite number) between same pair of vertices, the resulting graph is called a multigraph. In particular, if G is a simple graph then for any λ ≥ 1, G (λ) and λG respectively denote the multigraph with edge-multiplicity λ and the disjoint union of λ copies of G. By a decomposition of a graph G, we mean a list of edge-disjoint subgraphs H1,…, H푘, of G whose union is G (ignoring isolated vertices). When each subgraph in a decomposition is isomorphic to H, we say that G has an H-decomposition. It is easily seen that ∑ e(Hi) = e(G) is one of the obvious necessary condition for the existence of a {H1,H2,…,H푘} – decomposition of G. A {H1,H2} – decomposition of G is a decomposition of G into copies of H1 and H2 using at least one of each. If G has a {H1,H2}-decomposition, we say that G is {H1,H2}-decomposable. The problem of H-decomposition of K푛(λ) is the well-known Alspach’s conjecture [6] when H is any set of cycles of length at most 푛 satisfying the necessary sum conditions and 2 | λ (푛-1) . For the case λ =1, Alspach conjecture is also stated for even values of 푛 , where in this case the cycles should decompose K푛 minus a one-factor. There are many related results, but only special cases of this conjecture are solved completely. When H is a set of paths, in this case the problem of H- decomposition has been investigated by Tarsi [19] who showed that if (푛−1) λ is even and H is any set of paths of length at most 푛 − 3 satisfying the necessary sum condition, then K푛(λ) has an H-decomposition. The problem of H-decomposition of K푚,푛(λ ) has been investigated by Truszczynski [20] when 푚 and 푛 are even and H is any set of the paths with some constraints on length satisfying the necessary sum condition. It is natural to consider the problem of H- decomposition of K푛, where H is a combination of paths, cycles, and some other subgraphs. We will restrict our attention to H which is any set of paths and cycles satisfying the necessary sum condition. There are several similarly known results as follows. A graph-pair of order 푡 consists of two non-isomorphic graphs G and H on 푡 non-isolated vertices for which G ∪ H is isomorphic to K푡. Study on {H1,H2}-decomposition of graphs is not new. Abueida, Daven, and Rob- lee [1,3] completely determined the values of 푛 for which K푛(λ ) admits the {H1,H2}-decomposition such that H1 ∪ H2 ≌ K푡, when λ ≥ 1 and |V (H1)| = |V (H2)| = 푡, where 푡 ∈ {4, 5}. Abueida and Daven [2] proved that there exists a {K푘, S푘+1}- decomposition of K푛 for 푘 ≥ 3 and 푛 ≡ 0, 1 (mod 푘). Abueida and O’Neil [4] proved that for 푘 ∈ {3, 4, 5}, the {C푘, S푘}-decomposition of K푛(λ) exists, whenever 푛 ≥ 푘 + 1 except for the ordered triples (푘,푛, λ) ∈ {(3, 4, 1), (4, 5, 1), (5, 6, 1), (5, 6, 2), (5, 6, 4), (5, 7, 1), (5, 8, 1)}. Abueida and Daven [2] obtained necessary and sufficient conditions for the {C4, (2K2)}- decomposition of the Cartesian product and tensor product of paths, cycles, and complete graphs. Shyu [14] obtained a necessary and sufficient condition for the existence of a {P5, C4}-decomposition of K푛. Shyu [15] proved that K푛 has a {P4, S4}-decomposition if and only if 푛 ≥ 6 and 3(푝+푞)= . Also he proved that K푛 has a {P푘, S푘}-decomposition with a restriction 푝 ≥ 푘/2, when 푘 even (resp.,

푝 ≥ 푘, when 푘 odd). Shyu [16] obtained a necessary and sufficient condition for the existence of a {P4,K3}- decomposition of K푛. Shyu [17] proved that K푛 has a {C4, S5} - decomposition if and only if 4(푝+푞) = , DOI: 10.9790/5728-11543133 www.iosrjournals.org 31 | Page Decomposition of Line Graph into Paths and Cycles

푞 ≠ 1, when 푛 is odd and 푞 ≥ max { 3, }, when 푛 is even. Shyu [18] proved that K푚,푛 has a {P푘, S푘}- decomposition for some 푚 and 푛 and also obtained some necessary and sufficient condition for the existence of a {P4, S4}-decomposition of K푚,푛. Sarvate and Zhang [13] obtained necessary and sufficient conditions for the existence of a {푝P3, 푞K3}-decomposition of K푛(λ), when 푝 = 푞. Chou et al. [8] proved that for a given triple (푝,푞,푟) of nonnegative integers, G decompose into 푝 copies of C4, 푞 copies of C6, and 푟 copies of C8 such that 4푝+6푞+8푟 = |E(G)| in the following two cases: (a) G = K푚,푛 with 푚 and 푛 both even and greater than four (b) G = K푛,푛 − I, where 푛 is odd. Chou and Fu [9] proved that the existence of a {C4,C2푡}-decomposition of K2푢,2푣, where 푡/2 ≤ 푢,푣 < 푡, when 푡 even (resp., (푡 + 1)/2 ≤ 푢,푣 ≤ (3푡 − 1)/2, when 푡 odd) implies such decomposition in K2m,2n, where 푚,푛 ≥ 푡 (resp., 푚,푛 ≥ (3푡 + 1)/2). Lee and Chu [10,11] obtained a necessary and sufficient condition for the existence of a {P푘, S푘}-decomposition of K푛,푛 and K푚,푛. Lee and Lin [12] obtained a necessary and sufficient condition for the existence of a {C푘, S푘+1}- decomposition of K푛,푛 − I. Abueida and Lian [5] obtained necessary and sufficient conditions for the existence of a {C푘, S푘+1}-decomposition of K푛 for some 푛. In this paper we investigate decompositions of line graph of K푛 into 푝 copies of P5 and 푞 copies of C4 for all possible values of 푝 ≥ 0 and 푞 ≥ 0.

II. {P5,C4}-decomposition of L(K푛) In this section, we investigate the existence of {P5,C4}-decomposition of L(Kn). Note that, the line graph of C4 is C4.

III. Construction: Let and be two cycles of length 4, where =(푎푏푐푑푎) and =(푤푥푦푧푤). If 푣 is a only common vertex of and , say 푐 = 푦 = 푣, then we have two paths of length 4 as follows: 푎푏푣푧푤,푤푥푣푑푎.

Lemma 2.1. There exists a {P5, C4}-decomposition of K4,4.

Proof . Let V (K4,4) = {푥1, 푥2, 푥3, 푥 4} ∪ {푦1, 푦2, 푦3, 푦4}. We exhibit the {P5,C4}-decomposition of K4,4 as follows:

(1) 푝 = 0 and 푞 = 4. The required cycles are (푥1푦1푥2푦2푥1), (푥1푦3푥2푦4푥1), (푥3푦1푥4푦2푥3), (푥3푦3푥4푦4푥3). (2) 푝 = 2 and 푞 = 2. The required paths and cycles are 푦1푥1푦2 푥2푦4, 푦4푥1푦3푥2푦1 and (푥3푦1푥4푦2푥3), (푥3푦3푥4푦4푥3). (3) 푝 = 3 and 푞 = 1. The required paths and cycles are 푥2푦2푥3푦1푥1, 푥1푦3푥2푦1푥4, 푥4푦2푥1푦4푥2 and (푥3푦3푥4푦4푥3). (4) p = 4 and q = 0. The required paths are 푦1푥1푦2푥2푦4, 푦4푥1푦3푥2푦1, 푦3푥3푦4푥4푦2, 푦2 푥3푦1푥4푦.3 .

Lemma 2.2. There exists a {P5,C4}-decomposition of the complete bipartite graph K8,8푚 for all 푚 ≥ 1.

Proof. Let V (K8,8푚) = (X, Y ) with X = {푥1, . . . , 푥8} Y = {푦1, . . . , 푦8m}. Partition (X, Y ) into 4-subsets ( , ) such that = X, = Y. Then G[ , ] ≅ K4,4. Thus K8,8푚 = 4푚(K4,4). By Lemma 2.1, the graph K4,4 has a {P5,C4}-decomposition. Hence the graph K8,8푚 has the desired decomposition.

Lemma 2.3. There exists a {P5,C4}-decomposition of the graph K8푚 − F for all 푚 ≥ 1, where F is a 1-factor of K8푚.

Proof. Let 8푚 = 4푘, where 푘 is a positive integer and V (K4푘 − I) = {푥0, . . . , 푥4k−1} = { }, where I = {푥2i 푥2i+1 : 0 ≤ 푖 ≤ 2푘−1} and A푖 = {푥2i, 푥2i+1}, 0 ≤ 푖 ≤ 2푘−1. We obtain a new graph G from K4푘 −I, by identifying each set A푖 with a single vertex a푖, join two vertices a푖 and a푗 by an edge if the corresponding sets A푖 and A푗 form a K|A푖|,|A푗| in K4푘. Then the new graph G =K2푘. The graph K2푘 has a hamilton path decomposition {G0, . . . ,G푘}, where each G푖, 0 ≤ 푖 ≤ 푘 is a hamilton path. Each G푖 decomposes some copies of P3 or P4. When we go back to K4푘 − I, each P3 (resp., P4) of G푖, 0 ≤ 푖 ≤ 푘 will gives rise to 2C4 or 2P5 (resp., {2P5, 1C4} or 3P5) in K4푘 − I. Hence the graph K푚 − I has the desired decomposition.

Lemma 2.4. There exit a {P5, C4} – decomposition of the graph L(K9).

DOI: 10.9790/5728-11543133 www.iosrjournals.org 32 | Page Decomposition of Line Graph into Paths and Cycles

Proof. For 0 ≤ 푖 ≤ 8, by Lemma 2.3 there exit a {P5,C4}-decomposition G푖 ≅ K8 – F푖, where G푖 defined on the vertex set {{푖,푗}|0≤ 푗 ≤ 8, 푗 ≠ 푖} and where F푖 = {{{ 푖 , 1+푖},{ 푖,2+푖}, {푖 ,3+푖},{ 푖,7+푖},},{{푖, 4+푖},{푖, 8+푖}, {푖, 5+푖},{푖,6+푖}}}, reducing all sums modulo 9.Then a {P5,C4}-decomposition of L(K9) on the vertex set {{푖,푗}|{푖,푗} ⊆ {0,1,....,8}} follows from the partition of the edges of into P5 or C4, {({푗,푗+1},{1+푗,5+ 푗 }, {5+ 푗,2+ 푗 },{2+ 푗 , 푗 }) | 0 ≤ 푗 ≤ 8}, again reducing all sums modulo 9.

Theorem 2.1. If 푛 ≡ 1 (mod 8) then there exists a {P5,C4} – decomposition of L(K푛). Proof. The result is true for 푛 = 9, so we proceed by induction on 푛. Let 푛 = 8푚 + 1, 푚 ≥ 2, and let the vertex set of k푛 be {∞} ⋃ {1,...,8푚}. L(K푛) can be partitioned into the following edge-disjoint subgraphs: 1. L(K9), K9 being defined on the vertex set {∞} ⋃ {1,…,8}; 2. L(K8푚-7), K8푚-7 being defined on the vertex set {∞} ⋃ {9,…,8푚}; 3. For 1 ≤ 푖 ≤ 8, G푖 ≅ K8푚-8 – F푖 on the vertex set {{ 푖 , 푗 }| 9 ≤ 푗 ≤ 8푚}, where F푖 ={{ 푖 , 2푘-1}, { 푖 , 2푘} | 5 ≤ 푘 ≤ 4푚; 4. For 9 ≤ 푗 ≤ 8푚, G푖 ≅ K8 – F푗 on the vertex set {{ 푖,푗 }| 1 ≤ 푖 ≤ 8}, where F푗 = {{2푘-1,푗},{2푘,푗}| 1 ≤ 푘 ≤ 4; 5. ( ) ⋃ ( );

6. K8,8푚-8 with b-partition {{∞ , 푖}|1 ≤ 푖 ≤ 8} and {{∞, 푗}| 9 ≤ 푗 ≤ 8푚; 7. For 1 ≤ 푖 ≤ 8, H푖 ≅ K8,8푚-8 with bipartition {{푖,푘}| 푘 ∈{∞,1,2,…,8} \{푖}} and {{푖,푗}| 9 ≤ 푗 ≤ 8푚}; and 8. For 9 ≤ 푗 ≤ 8푚, H푗 ≅ K8,8푚-8 with bi-partition {{푖,푗}|1 ≤ 푖 ≤ 8} and {{푘 ,푗}| 푘 ∈ {∞,9,10,…,8푚}\ {푗}}. The result now follows, since there exist {P5,C4}-decomposition of graphs defined in (1) and (2) by induction and Lemma 2.4, (3) and (4) by Lemma 2.3, (5) since these edges form vertex-disjoint C4, and (6),(7) and (8) by Lemma 2.2. Hence the graph L(K푛) has the desired decomposition.

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DOI: 10.9790/5728-11543133 www.iosrjournals.org 33 | Page