Circular Chromatic Numbers and Fractional Chromatic Numbers of Distance Graphs with Distance Sets Missing An Interval

Abstract

Given positive integers $m,k,$ and $t$. Let $D_{m,[k,k+i]}=\{1,2,\dots ,m\}–\{k,k+1,\dots ,k+i\}$. The distance graph $G(\mathbb{Z},D_{m,[k,k+i]})$ has vertex set all integers $\mathbb{Z}$ and edges connecting $j$ and $j^{\prime }$ whenever $|j-j^{\prime }|\in D_{m,[k,k+i]}$. The fractional chromatic number, the chromatic number, and the circular chromatic number of $G(\mathbb{Z},D_{m,k,i})$ are denoted by ${\chi }_f(\mathbb{Z},D_{m[k,k+i]}),\chi (\mathbb{Z},D_{m,[k,k+i]}),$ and ${\chi }_c(\mathbb{Z},D_{m,[k,k+i]})$, respectively. For $i=0$, we simply denote $D_{m,[k,k+0]}$ by $D_{m,k}$. $X(\mathbb{Z},D_{m,k})$ was studied by Eggleton, Erdős and Skilton [5], Kemnitz and Kolberg [8], and Liu [9], and was completely solved by Chang, Liu and Zhu [1] who also determined ${\chi }_c(\mathbb{Z},D_{m,k})$ for any $m$ and $k$. The value of ${\chi }_c(\mathbb{Z},D_{m,k})$ was studied by Chang, Huang and Zhu [2] who finally determined ${\chi }_c(\mathbb{Z},D_{m,k})$ for any $m$ and $k$. This paper extends the study of $G(\mathbb{Z},D_{m,[k,k+i]})$ to values $i$ with $1\le i\le k-1$. We completely determine ${\chi }_f(\mathbb{Z},D_{m,[k,k+i]})$ and $\chi (\mathbb{Z},D_{m,k,i})$ for any $m$ and $k$ with $1\le i\le k-1$. However, for ${\chi }_c(\mathbb{Z},D_{m,[k,k+i]})$, only some special cases are determined.