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arXiv 1506.04401math.CO
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最大平均度小于小于4的图的平方着色

Coloring the squares of graphs whose maximum average degrees are less than 4

Seog-Jin Kim, Boram Park

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摘要：

图$G$的正方形$G^2$是定义在$V(G)$上的图，使得如果$G$中的$u$和$v$之间的距离至多为2，则$G^2$中的两个顶点$u$和$v$相邻。$G$的{ \em最大平均度} $mad (G)$是$G$的子图的平均度中的最大值。已知在\cite{BLP-14-JGT}中不存在常数$C$，使得具有$mad(G)< 4$的每个图$G$具有$χ(G^2) \leq Δ(G) + C$。Charpentier \cite{Charpentier14}证明了存在一个整数$D$，使得每个具有$Δ(G)\ge D$和$mad(G)<4$的图$G$都有$χ(G^2) \leq 2 Δ(G)$ . \cite{BLP-DM}的最新结果表明，如果$mad(G) < 4 -\frac{1}{c}$与$Δ(G) \geq 40c -16$联用，则$χ(G^2) \leq 2 Δ(G)$。本文证明了对于$c\ge 2$，如果$mad(G) < 4 - \frac{1}{c}$和$Δ(G) \geq 14c-7$，则有$χ_\ell(G^2) \leq 2 Δ(G)$，改进了\cite{BLP-DM}的结果.我们还证明了对任意整数$D$，存在一个具有$Δ(G)\ge D$的图$G$，使得$mad(G)<4$，和$χ(G^2) \geq 2Δ(G) +2$，从而反驳了Charpentier猜想.另外，我们给出了Charpentier在\cite{Charpentier14}中的另一个猜想的反例，指出对于每个整数$k\ge 3$，存在一个整数$D_k$使得每个具有$mad(G)<2k$和$Δ(G)\ge D_k$的图$G$有$χ(G^2) \leq kΔ(G) -k$ .

英文摘要：

The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. The {\em maximum average degree} of $G$, $mad (G)$, is the maximum among the average degrees of the subgraphs of $G$. It is known in \cite{BLP-14-JGT} that there is no constant $C$ such that every graph $G$ with $mad(G)< 4$ has $χ(G^2) \leq Δ(G) + C$. Charpentier \cite{Charpentier14} conjectured that there exists an integer $D$ such that every graph $G$ with $Δ(G)\ge D$ and $mad(G)<4$ has $χ(G^2) \leq 2 Δ(G)$. Recent result in \cite{BLP-DM} implies that $χ(G^2) \leq 2 Δ(G)$ if $mad(G) < 4 -\frac{1}{c}$ with $Δ(G) \geq 40c -16$. In this paper, we show for $c\ge 2$, if $mad(G) < 4 - \frac{1}{c}$ and $Δ(G) \geq 14c-7$, then $χ_\ell(G^2) \leq 2 Δ(G)$, which improves the result in \cite{BLP-DM}. We also show that for every integer $D$, there is a graph $G$ with $Δ(G)\ge D$ such that $mad(G)<4$, and $χ(G^2) \geq 2Δ(G) +2$, which disproves Charpentier's conjecture. In addition, we give counterexamples to Charpentier's another conjecture in \cite{Charpentier14}, stating that for every integer $k\ge 3$, there is an integer $D_k$ such that every graph $G$ with $mad(G)<2k$ and $Δ(G)\ge D_k$ has $χ(G^2) \leq kΔ(G) -k$.

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