Odd Induced Subgraphs in Graphs of Maximum Degree Four

TL;DR

This paper proves that in graphs with maximum degree four and no isolated vertices, the largest guaranteed proportion of vertices in an odd induced subgraph is exactly 2/7, confirming a longstanding conjecture.

Contribution

The paper establishes that the bound c=2/7 is tight for graphs with maximum degree four, advancing understanding of odd induced subgraphs in bounded degree graphs.

Findings

The maximum degree four graphs without isolated vertices have an odd induced subgraph covering at least 2/7 of the vertices.

The bound c=2/7 is proven to be tight for this class of graphs.

Supports the conjecture that 2/7 is the largest possible constant c.

Abstract

A graph is called odd if all of its vertex degrees are odd. A long-standing conjecture asked whether there exists a positive constant $c$ such that every $n$-vertex graph without isolated vertices contains an odd induced subgraph on at least $cn$ vertices. In 2022, Ferber and Krivelevich resolved this conjecture affirmatively with $c=10^{-4}$. A natural question is to determine the largest possible constant $c$. In 1994, Caro remarked that if $2/7$ is a valid value for $c$, then it is the largest possible one. To the best of our knowledge, the bound $c\ge 2/7$ has not been improved. Previous research has established tight bounds for specific graph classes -- for instance, $c = 2/5$ for graphs with maximum degree at most $3$ and without isolated vertices. In this paper, we prove that $c=2/7$ is the tight bound for graphs with maximum degree at most $4$ and without isolated vertices. Our result provides some support for $2/7$ being the largest value of $c$.