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arXiv · 1103.1521

On The b-Chromatic Number of Regular Graphs Without 4-Cycle

Saeed Shaebani
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Abstract

The b-chromatic number of a graph $G$, denoted by $ϕ(G)$, is the largest integer $k$ that $G$ admits a proper $k$-coloring such that each color class has a vertex that is adjacent to at least one vertex in each of the other color classes. We prove that for each $d$-regular graph $G$ which contains no 4-cycle, $ϕ(G)\geq\lfloor\frac{d+3}{2}\rfloor$ and if $G$ has a triangle, then $ϕ(G)\geq\lfloor\frac{d+4}{2}\rfloor$. Also, if $G$ is a $d$-regular graph which contains no 4-cycle and $diam(G)\geq6$, then $ϕ(G)=d+1$. Finally, we show that for any $d$-regular graph $G$ which does not contain 4-cycle and $κ(G)\leq\frac{d+1}{2}$, $ϕ(G)=d+1$.

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Saeed Shaebani. 2011-03-08. On The b-Chromatic Number of Regular Graphs Without 4-Cycle. https://arxiv.org/abs/1103.1521

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