Extremal graphs for disjoint union of vertex-critical graphs
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909503990005760 |
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| author | Zhang, Wenqian |
| author_facet | Zhang, Wenqian |
| contents | 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extremal graphs for disjoint union of vertex-critical graphs Zhang, Wenqian Combinatorics 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}. |
| title | Extremal graphs for disjoint union of vertex-critical graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.15456 |