Extremal graphs for disjoint union of vertex-critical graphs

TL;DR

This paper characterizes extremal graphs avoiding a disjoint union of vertex-critical graphs with the same chromatic number, solving a conjecture related to odd wheels and extremal graph theory.

Contribution

It provides a complete characterization of extremal graphs for disjoint unions of vertex-critical graphs under a proper order, addressing a conjecture by Xiao and Zamora.

Findings

Characterization of extremal graphs for disjoint unions of vertex-critical graphs.

Resolution of a conjecture on extremal problems for odd wheels.

Extension of extremal graph theory to ordered disjoint unions.

Abstract

For a graph $F$, let ${\rm EX}(n,F)$ be the set of $F$-free graphs of order $n$ with the maximum number of edges. The graph $F$ is called vertex-critical, if the deletion of its some vertex induces a graph with smaller chromatic number. For example, an odd wheel (obtained by connecting a vertex to a cycle of even length) is a vertex-critical graph with chromatic number 3. For $h\geq2$, let $F_{1},F_{2},...,F_{h}$ be vertex-critical graphs with the same chromatic number. Let $\cup_{1\leq i\leq h}F_{i}$ be the disjoint union of them. In this paper, we characterize the graphs in ${\rm EX}(n,\cup_{1\leq i\leq h}F_{i})$, when there is a proper order among the graphs $F_{1},F_{2},...,F_{h}$. This solves a conjecture (on extremal problem for disjoint union of odd wheels) proposed by Xiao and Zamora \cite{XZ}.