Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Path decompositions of Eulerian graphs

Yanan Chu, Yan Wang

TL;DR

The paper addresses Gallai's conjecture on path decompositions, focusing on Eulerian graphs with many even-degree vertices. It introduces the concept of a triangle removal set and derives a bound $p(G) \le \frac{3n}{5}$ when a triangle-removal structure exists and, crucially, when the distance between any two triangles is at least $3$, by reducing to triangle-free cases and applying the CFZ bound along with new decomposition lemmas. The main result shows that for Eulerian graphs on $n \ge 4$ with the triangle-distance condition, $p(G) \le \frac{3n}{5}$, extending understanding of Gallai-type bounds in dense-even-degree settings. This provides a structured decomposition approach via triangle management and cycle-path lemmas, contributing a concrete bound in a challenging regime of Gallai's conjecture.

Abstract

Gallai's conjecture asserts that every connected graph on $n$ vertices can be decomposed into $\frac{n+1}{2}$ paths. For general graphs (possibly disconnected), it was proved that every graph on $n$ vertices can be decomposed into $\frac{2n}{3}$ paths. This is also best possible (consider the graphs consisting of vertex-disjoint triangles). Lovász showed that every $n$-vertex graph with at most one vertex of even degree can be decomposed into $\frac{n}{2}$ paths. However, Gallai's conjecture is difficult for graphs with many vertices of even degrees. Favaron and Kouider verified Gallai's conjecture for all Eulerian graphs with maximum degree at most $4$. In this paper, we show if $G$ is an Eulerian graph on $n \ge 4$ vertices and the distance between any two triangles in $G$ is at least $3$, then $G$ can be decomposed into at most $\frac{3n}{5}$ paths.

Path decompositions of Eulerian graphs

TL;DR

The paper addresses Gallai's conjecture on path decompositions, focusing on Eulerian graphs with many even-degree vertices. It introduces the concept of a triangle removal set and derives a bound when a triangle-removal structure exists and, crucially, when the distance between any two triangles is at least , by reducing to triangle-free cases and applying the CFZ bound along with new decomposition lemmas. The main result shows that for Eulerian graphs on with the triangle-distance condition, , extending understanding of Gallai-type bounds in dense-even-degree settings. This provides a structured decomposition approach via triangle management and cycle-path lemmas, contributing a concrete bound in a challenging regime of Gallai's conjecture.

Abstract

Gallai's conjecture asserts that every connected graph on vertices can be decomposed into paths. For general graphs (possibly disconnected), it was proved that every graph on vertices can be decomposed into paths. This is also best possible (consider the graphs consisting of vertex-disjoint triangles). Lovász showed that every -vertex graph with at most one vertex of even degree can be decomposed into paths. However, Gallai's conjecture is difficult for graphs with many vertices of even degrees. Favaron and Kouider verified Gallai's conjecture for all Eulerian graphs with maximum degree at most . In this paper, we show if is an Eulerian graph on vertices and the distance between any two triangles in is at least , then can be decomposed into at most paths.
Paper Structure (3 sections, 8 theorems, 6 equations, 2 figures)

This paper contains 3 sections, 8 theorems, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Decomposition lemmas
  3. Proof of main theorems

Key Result

Theorem 1.2

Let $G$ be a graph (possibly disconnected) on $n$ vertices. If $G$ contains at most one vertex of even degree, then $p(G)\leq \frac{n}{2}$.

Figures (2)

  • Figure 1: The exceptional graph with $|V(C)|=5$ and $|V(P)\cap V(C)|=5$ ($C$ is depicted by thick solid lines; $P$ is depicted by thin lines, with dashed lines used to represent its subpaths).
  • Figure 2: $C_1$, $C_2$ and $P$ are in red, blue and black, respectively.

Theorems & Definitions (14)

  • Conjecture 1.1
  • Theorem 1.2: Lovász L
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1: Lemma 2.1 in CFL
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 4 more