Anti-Ramsey Number for Suspension of Edge-Critical Graphs

An edge-colored graph is called a rainbow graph if all its edges have distinct colors. The $\mathit{\text{anti-Ramsey}}\mathit{\text{number}}$, denoted by $\text{ar}(n,F),$ 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 $\mathit{\text{Tur}}\acute{\mathit{\text{a}}}\mathit{\text{n}}\mathit{\text{number}}$, denoted by $\text{ex}(n,F),$ 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 $\mathit{\text{suspension}}$ $\mathcal{H}+v$ 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 ${\mathcal{H}}_{k+1}=\{H_1,H_2,\dots ,H_{k+1}\}+v$ satisfying $H_1,H_2,\dots ,H_{k+1}$ are pairwise vertex-disjoint edge-critical graphs, and $\chi (H_i)=r$ for $i=1,2,\dots ,k+1$.In this paper, we determine $\text{ar}(n,{\mathcal{H}}_{k+1})$ 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.