Forbidding edge-critical graphs as trace in uniform hypergraphs

TL;DR

This paper extends Turán-type extremal results for hypergraph traces to include edge-critical graphs, establishing that large uniform Turán hypergraphs uniquely maximize edges without containing certain trace graphs.

Contribution

It generalizes previous results to all edge-critical graphs with chromatic number s+1, showing Turán hypergraphs are extremal for these cases.

Findings

For large n, the extremal hypergraph avoiding a trace of an edge-critical graph is the Turán hypergraph.

The result applies when the chromatic number s+1 satisfies s ≥ r ≥ 3.

The Turán hypergraph uniquely maximizes edges among hypergraphs avoiding the trace of the given graph.

Abstract

We say a hypergraph $\mathcal{H}$ contains a graph $G$ as trace if there exists a vertex subset $S \subseteq V(\mathcal{H})$ such that $|S| = V(G)$ and $\{e \cap S \mid e \in E(\mathcal{H})\}$ contains $G$ as a subgraph. We use $\mathrm{ex}(n, Tr_r(G))$ to denote the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices not containing $G$ as trace. The study of Tur\'an numbers for traces was initiated by Mubayi and Zhao~(2017) who studied $\mathrm{ex}(n, Tr_r(K_{s+1}))$ where $K_{s+1}$ is a clique on $s+1$ vertices and conjectured the exact value of $\mathrm{ex}(n, Tr_r(K_{s+1}))$. When $r \le s$, the conjecture was covered by a result of Pikhurko~(2013) who gave the exact value of Tur\'an numbers for expanded cliques. Then Gerbner and Picollelli~(2023) gave the exact value for book graphs~($K_{1,1,t}$, the complete tripartite graph with two parts of size one and one part of size $t \ge 2$). We say $G$ is edge-critical if there exists an edge $e \in E(G)$ such that $\chi(G - e) < \chi(G)$ where $\chi(G)$ is the chromatic number of $G$. The definition of edge-critical was given by Simonovits~(1974), who proved that for an edge-critical graph $G$ with $\chi(G) = s+1 \ge 3$, the Tur\'an graph $T(n,s)$ is the unique extremal graph for $ex(n,G)$ as $n$ is sufficiently large. In this paper, we further generalize the results of Gerbner and Picollelli~(2023) to edge-critical graphs. More precisely, we prove that for an edge-critical graph $G$ with $\chi(G) = s+1$, when $s \ge r \ge 3$ and $n$ is sufficiently large, the $r$-uniform Tur\'an graph $T_r(n,s)$ is the unique extremal hypergraph.