Extremal graphs for disjoint union of vertex-critical graphs
Wenqian Zhang
TL;DR
The paper determines the Tur\'an-type extremal numbers for disjoint unions of vertex-critical graphs with the same chromatic number, resolving a conjecture by Xiao and Zamora for disjoint unions of odd wheels. It introduces the notion of a properly ordered family and proves that for large $n$ the extremal number ex$(n,\\cup_{1\\le i\\le h} F_i)$ equals the maximum over $\\ell$ of $\\binom{\\ell-1}{2}+(\\ell-1)(n-\\ell+1)+\\mathrm{ex}(n-\\ell+1,F_{\\ell})$, with extremals of the form $K_{\\ell-1}\\prod H$ where $H\\in{\\rm EX}(n-\\ell+1,F_{\\ell})$. The approach leverages Tur\'an-type stability and a constructive embedding argument to both bound and characterize extremal graphs. As a consequence, the conjecture for disjoint unions of odd wheels is settled and a unified framework for ex-values of unions of vertex-critical graphs is obtained.
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}.