The Fractional Chromatic Number of Generalized Cones over Graphs

  • Jialu Zhu
  • Xuding Zhu


  For a graph $G$ and a positive integer $n$, the $n$th cone over $G$ is obtained from the direct product $G \times P_n$ of $G$ and a path $P_n=(0,1,\ldots, n)$, by adding a copy of $G$ on $V(G) \times \{0\}$, and identifying $V(G) \times \{n\}$ into a single vertex $\star$. Assume $G$ and $H$ are graphs, and $h: V(H) \to \mathbb{N}$ is a mapping which assigns to each vertex $v$ of $H$ a positive integer. For each vertex $v$ of $H$, let $\Delta_{h(v)}(G,v)$ be a copy of the $h(v)$-th cone over $G$, with vertex set $V(\Delta_{h(v)}(G)) \times \{v\}$. The $(H,h)$-cone over $G$ is the graph obtained from the disjoint union of $\{\Delta_{h(v)}(G, v) : v\in V(H)\}$ by identifying $\{((x,0),v): v \in V(H)\}$ into a single vertex $(x,0)$ for each $x \in V(G)$, and adding edges $\{(\star, v) (\star, v'): vv' \in E(H)\}$. When $h(v)=n$ is a constant mapping, then $\Delta_{H,h}(G)$ is denoted by $\Delta_{H,n}(G)$. In this paper, we determines the fractional chromatic number of $\Delta_{H,n}(G)$ for all $G, H$ with $\chi_f(H)\le \chi_f(G)$.

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