Sufficient Condition for the Existence of Three Disjoint Theta Graphs

A theta graph is the union of three internally disjoint paths that have the same two distinct end vertices. We show that every graph of order n ≥ 12 and size at least ⌊ 11n−18 2 ⌋ contains three disjoint theta graphs. As a corollary, every graph of order n ≥ 12 and size at least ⌊ 11n−18 2 ⌋ contains three disjoint cycles of even length. 1. Terminology and introduction In this paper, we only consider finite undirected graphs, without loops or multiple edges. We use [1] for the notation and terminology not defined here. A theta graph is the union of three internally disjoint paths that have the same two distinct end vertices. Let n be a positive integer, let Kn denote the complete graph of order n and K 4 be the graph obtained by removing exactly one edge from K4. For a graph G, we denote its vertex set, edge set, minimum degree by V (G), E(G) and δ(G), respectively. The order and size of a graph G, are defined by |V (G)| and |E(G)|, respectively. A set of subgraphs is said to be vertex-disjoint or independent, if no two of them have any common vertex in G, and we use disjoint to stand for vertex-disjoint throughout this paper. If u is a vertex of G and H is either a subgraph of G or a subset of V (G), we define NH(u) to be the set of neighbors of u contained in H , and dH(u) = |NH(u)|. For a subset U of V (G), G[U ] denotes the subgraph of G induced by U . In particular, we often use [U ] to stand for G[U ]. If S is a set of subgraphs of G, we write G ⊇ S, it means that S is isomorphic to a subgraph of G, in particular, we use mS to represent a set of m vertex-disjoint copies of S. When S = {x1, x2, . . . , xt}, we may also use [x1, x2, . . . , xt] to denote [{x1, x2, . . . , xt}]. Let V1, V2 be two disjoint subsets or subgraphs of G, we use E(V1, V2) to denote the set of edges in G with one end-vertex in V1, while the other in V2, for simplicity, let E(x, V2) stand for E({x}, V2), E(V1, x) Received February 5, 2014; Revised July 10, 2014. 2010 Mathematics Subject Classification. 05C35, 05C70.