边临界图的悬浮图的反拉姆齐数

中文摘要 AI 辅助

边着色图若所有边颜色互不相同则称为彩虹图。对于固定图$F$和正整数$n$，反拉姆齐数记为$\text{ar}(n,F)$，是完全图$K_n$的边着色中不含$F$的彩虹副本时使用的最大颜色数；图$F$和$n$的Turán数记为$\text{ex}(n,F)$，是不含$F$作为子图的$n$顶点图的最大边数。对于顶点$v$和图的多重集$\boldsymbol{\textit{H}}$，$\boldsymbol{\textit{H}}$的悬浮图$\boldsymbol{\textit{H}}+v$是将顶点$v$与$\boldsymbol{\textit{H}}$中每个图$H$的所有顶点相连得到的图。设整数$k\boldsymbol{\text{≥}}1$、$r\boldsymbol{\text{≥}}2$固定，且$\boldsymbol{\textit{H}}_{k+1}=\boldsymbol{\textit{\text{\{}}}}H_1,H_2,\boldsymbol{\text{\ldots}},H_{k+1}\boldsymbol{\textit{\text{\}}}}+v$满足$H_1,H_2,\boldsymbol{\text{\ldots}},H_{k+1}$是两两顶点不相交的边临界图，且对$i=1,2,\boldsymbol{\text{\ldots}},k+1$有$\boldsymbol{\text{\text{χ}}}(H_i)=r$。本文针对$k\boldsymbol{\text{≥}}1$、$r\boldsymbol{\text{≥}}2$且足够大的$n$，确定了$\text{ar}(n,\boldsymbol{\textit{H}}_{k+1})$的值，该结果统一并推广了Liu等人（arXiv:2411.08475）关于友谊图的结果，以及Lu等人（arXiv:2507.13165）关于相交团的结果。

英文摘要

An edge-colored graph is called a rainbow graph if all its edges have distinct colors. The \textit{anti-Ramsey number}, denoted by $\ar(n,F),$ for a fixed graph $F$ and a positive integer $n$, is the maximum number of colors used in an edge-coloring of the complete graph $K_n$ that contains no rainbow copy of $F$. Meanwhile, the \textit{Turán number}, denoted by $\ex(n,F),$ for graph $F$ and $n$, is the maximum number of edges in an $n$-vertex graph that does not contain $F$ as a subgraph. For a vertex $v$ and a multiset $\mathcal{H}$ of graphs, the \textit{suspension} $\mathcal{H} + v$ of $\mathcal{H}$ is the graph obtained by connecting the vertex $v$ to all vertices of $H$ for each $H \in \mathcal{H}$. Let integers $k\ge 1$ and $r\ge 2$ be fixed, and suppose that $\mathcal{H}_{k+1}=\{H_1, H_2, \ldots, H_{k+1}\}+v$ satisfying $H_1, H_2, \ldots, H_{k+1}$ are pairwise vertex-disjoint edge-critical graphs, and $χ(H_i)=r$ for $i=1,2,\ldots, k+1$.In this paper, we determine $ \ar(n,\mathcal{H}_{k+1}) $ for $k\ge 1$, $r\ge 2$ and sufficiently large $n$. This result unifies and generalizes a result of Liu et al. (arXiv:2411.08475) concerning the friendship graph, and a result of Lu et al. (arXiv:2507.13165) on the intersecting cliques.

发表机构

• School of Mathematics and Statistics, and Hubei Key Lab–Math. Sci.,Central China Normal University（华中师范大学数学与统计学院、湖北省数学科学重点实验室）

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