Vertex Arboricity of Integer Distance Graph with a
Special Distance Set

Juan Liu 1 and Qinglin Yu 1, 2
1 Center for Combinatorics, LPMC, Nankai University, Tianjin 300071,
P. R. China
2 Department of Mathematics and Statistics, Thompson Rivers University,
Kamloops, BC, Canada


Abstract     Full Text  PPT

If $D$ is a subset of positive integer set $Z^{+}$ , then the distance graph $G(Z,D)$ is a graph with vertex set $Z$ and two vertices $x$ and $y$ are
adjacent if and only if $|x-y|\in D$ where the set $D$ is called the {\it distance set}. Recently, Zuo and Yu has investigated the vertex arboricity, $va(G(Z, D))$, of the distance graph $G(Z, D)$ and showed that $va(G(D_{m,2}))=\lceil\frac{m+1}{4} \rceil+1$ for $j\neq 7$ and $va(G(D_{m,2}))=\lceil\frac{m}{4}\rceil + 1$ or $\lceil\frac{m}{4}\rceil + 2$ for $j=7$, where $D_{m,2}=[1,m]\setminus \{2\}$ for any positive integer $m=8l+j$ ($l,j\in \mathbb{N})$. In this talk, we determine the value of $va(G(D_{m,2}))$ for the case $j = 7$.