Gallai’s Path Decomposition for 2-degenerate Graphs

Abstract

Gallai’s path decomposition conjecture states that if G is a connected graph on n vertices, then the edges of G can be decomposed into at most ⌈ n<inf>2</inf> ⌉ paths. A graph is said to be an odd semi-clique if it can be obtained from a clique on 2k + 1 vertices by deleting at most k − 1 edges. Bonamy and Perrett asked if the edges of every connected graph G on n vertices can be decomposed into at most ⌊ n<inf>2</inf> ⌋ paths unless G is an odd semi-clique. A graph G is said to be 2-degenerate if every subgraph of G has a vertex of degree at most 2. In this paper, we prove that the edges of any connected 2-degenerate graph G on n vertices can be decomposed into at most ⌊ n<inf>2</inf> ⌋ paths unless G is a triangle. © 2023 by the author(s)