Forbidding edge-critical graphs as trace in uniform hypergraphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910036405518336 |
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| author | Wang, Yichen Cheng, Xin Győri, Ervin Wang, Yuanpei Zhao, Xiamiao Zhou, Junpeng |
| author_facet | Wang, Yichen Cheng, Xin Győri, Ervin Wang, Yuanpei Zhao, Xiamiao Zhou, Junpeng |
| contents | 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 Turán 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 Turán 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 $χ(G - e) < χ(G)$ where $χ(G)$ 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 $χ(G) = s+1 \ge 3$, the Turán 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 $χ(G) = s+1$, when $s \ge r \ge 3$ and $n$ is sufficiently large, the $r$-uniform Turán graph $T_r(n,s)$ is the unique extremal hypergraph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09500 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Forbidding edge-critical graphs as trace in uniform hypergraphs Wang, Yichen Cheng, Xin Győri, Ervin Wang, Yuanpei Zhao, Xiamiao Zhou, Junpeng Combinatorics 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 Turán 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 Turán 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 $χ(G - e) < χ(G)$ where $χ(G)$ 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 $χ(G) = s+1 \ge 3$, the Turán 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 $χ(G) = s+1$, when $s \ge r \ge 3$ and $n$ is sufficiently large, the $r$-uniform Turán graph $T_r(n,s)$ is the unique extremal hypergraph. |
| title | Forbidding edge-critical graphs as trace in uniform hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2601.09500 |