Odd Induced Subgraphs in Graphs of Maximum Degree Four
Jiangdong Ai, Qiwen Guo, Gregory Gutin, Yiming Hao, Anders Yeo
TL;DR
The paper addresses the problem of determining the largest guaranteed size of an odd induced subgraph in $n$-vertex graphs without isolated vertices, focusing on graphs with maximum degree at most $4$. It proves that the optimal constant is $c=\\frac{2}{7}$, meaning every such graph has an odd induced subgraph of size at least $\\frac{2}{7}n$, and this bound is tight. The main method uses a minimal counterexample and extensive case analysis on minimum degree and local structure to derive contradictions, supplemented by a tightness example from a $K_7$ missing a Hamiltonian cycle. Together with the known sharp bound $c=\\frac{2}{5}$ for max degree $3$, this result completes the exact determination for graphs with maximum degree at most four and situates the finding in the broader context of related conjectures and structural graph theory.
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 $c$.