Gallai's Path Decomposition for 2-degenerate Graphs

TL;DR

We prove Gallai's path decomposition conjecture for the class of $2$-degenerate graphs by showing that a connected $2$-degenerate graph on $n$ vertices has an edge decomposition into at most $\lfloor \frac{n}{2} \rfloor$ paths, except when $G$ is a triangle. The proof combines a minimal counterexample argument with a constructive decomposition that handles triangle components via two paths $Q$ and $R$ and uses a $2$-degenerate vertex-removal ordering to control degrees and cut-vertices. A corollary extends the bound to (not necessarily connected) $2$-degenerate graphs provided none of the components is a triangle, and the work relates to broader results for graphs with bounded treewidth or triangle-free planar graphs, guiding future extension to $3$-degenerate graphs. This advances understanding of Gallai's conjecture by establishing the bound for a large structural class and outlining methods that may generalize to higher degeneracy.

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.

Figures (5)

• Figure 1: Three vertices of triangle $T_j$ intersect $P$. (a) The path $P$ is shown with a dashed line, while the edges of $T_j$ are shown in solid lines. (b) Decomposition of the edges of $T_j$ and $P$ into two paths $Q$ (thin line) and $R$ (bold line).
• Figure 2: Two vertices of triangle $T_j$ intersect $P$. (a) The path $P$ is shown with a dashed line, while the edges of $T_j$ are shown in solid lines. (b) Decomposition of the edges of $T_j$ and $P$ into two paths $Q$ (thin line) and $R$ (bold line).
• Figure 3: One vertex of triangle $T_j$ intersects $P$. (a) The path $P$ is shown with a dashed line, while the edges of $T_j$ are shown in solid lines. (b) Decomposition of the edges of $T_j$ and $P$ into two paths $Q$ (thin line) and $R$ (bold line).
• Figure 4: (a) Claim \ref{['lemma:deg1']} : Vertex $v$ cannot be a pendant vertex. (b) Claim \ref{['lemma:deg3cut']} : Vertex $x$ cannot be a cut vertex. Any vertex which is drawn as a circle has all its edges depicted in the figure. Rectangular vertices may have edges not depicted in the figure.
• Figure 5: Figure depicts scenarios if $x$ is not a cut vertex. Case \ref{['ncv1']}: $x$ has a degree $3$ neighbour $z$. Case \ref{['ncv2.1']}: $x$ has no degree $3$ neighbour and $d_G(y) = 3$. Case \ref{['ncv2.2']}: $x$ has no degree $3$ neighbour and $d_G(y) = 4$. Any vertex which is drawn as a circle has all its edges depicted in the figure. Rectangular vertices may have edges not depicted in the figure.

Theorems & Definitions (15)

• Theorem 1
• corollary 1
• definition 1: Vertex removal order
• lemma 1
• proof
• Claim 1
• proof
• proof
• proof
• Claim 2