Catlin's conjecture and maximum eulerian subgraph

TL;DR

This paper investigates Catlin's conjecture on the maximum edges of spanning Eulerian subgraphs in supereulerian graphs, proving it for specific classes and establishing bounds involving vertices of degree 2.

Contribution

The paper proves Catlin's conjecture for graphs without degree 3 vertices and for 5-regular graphs, and introduces a bound involving degree 2 vertices.

Findings

Catlin's conjecture holds for graphs with no degree 3 vertices.

Catlin's conjecture holds for 5-regular graphs.

A new lower bound involving degree 2 vertices is established.

Abstract

A graph $G=(V(G), E(G))$ is supereulerian if it has a spanning Eulerian subgraph. Let $\ell(G)$ be the maximum number of edges of spanning Eulerian subgraphs of a supereulerian graph $G$. In $1996$, Catlin conjectured that if $G$ is a supereulerian graph, then $\ell(G)\ge \frac{2}{3}|E(G)|$. But in $2004$, infinitely many counterexamples were found for this conjecture and it was shown that this conjecture holds for $r$-regular graphs when $r\neq 5$. In this paper we show that Catlin's Conjecture holds for graphs having no vertex with degree $3$ and also it holds for $5$-regular graphs. Moreover, if $G$ is a graph having no vertex with degree $3$, then $\ell(G)\ge \frac{2}{3}|E(G)|+ v_2(G)$, when $v_2(G)$ is the number of vertices of degree $2$.