Gallai's Path Decomposition for 2-degenerate Graphs

TL;DR

This paper proves that connected 2-degenerate graphs on n vertices can be decomposed into at most ⌊n/2⌋ paths, except for triangles, advancing understanding of Gallai's path decomposition conjecture.

Contribution

It establishes the path decomposition bound for 2-degenerate graphs, a class not previously fully addressed, and clarifies exceptions like triangles.

Findings

Connected 2-degenerate graphs can be decomposed into ⌊n/2⌋ paths.

The only exception is the triangle graph.

Supports Gallai's conjecture for a broader class of 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 $\lceil \frac{n }{2} \rceil$ 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 $\lfloor \frac{n}{2} \rfloor$ 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 $\lfloor \frac{n }{2} \rfloor$ paths unless $G$ is a triangle.