# 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.

## Key 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 $2$.

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Source: https://tomesphere.com/paper/1812.03893