Gallai's conjecture for 3-degenerated graphs

Abstract

Gallai's conjecture asserts that every connected graph of order n can be decomposed into ⌈n2⌉ paths. A graph G is k-degenerated if each subgraph admits a vertex with degree no more than k. In this paper, we characterize the graphs that contain a path through specified edges. As a result, we prove that a connected 3-degenerated graph of order n that is not isomorphic to K3 or K5− can be decomposed into ⌊n2⌋ paths, which extends three theorems of [2,3,12].