Path decompositions of triangle-free graphs

Abstract

Let G be a graph with n vertices. A path decomposition of G is a set of edge-disjoint paths containing all the edges of G. Let p(G) denote the minimum number of paths needed in a path decomposition of G. Gallai Conjecture asserts that if G is connected, then p(G)≤⌈n/2⌉. If G is allowed to be disconnected, then the upper bound ⌊34n⌋ for p(G) was obtained by Donald [7], which was improved to ⌊23n⌋ independently by Dean and Kouider [6] and Yan [14]. For graphs consisting of vertex-disjoint triangles, ⌊23n⌋ is reached and so this bound is tight. If triangles are forbidden in G, then p(G)≤⌊g+12gn⌋ can be derived from the result of Harding and McGuinness [11], where g denotes the girth of G. In this paper, we also focus on triangle-free graphs and prove that p(G)≤⌊3n/5⌋, which improves the above result with g=4.