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

The b-chromatic number of a graph $G$, denoted by $\phi(G)$, 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, $\phi(G)\geq\lfloor\frac{d+3}{2}\rfloor$ and if $G$ has a triangle, then $\phi(G)\geq\lfloor\frac{d+4}{2}\rfloor$. Also, if $G$ is a $d$-regular graph which contains no 4-cycle and $diam(G)\geq6$, then $\phi(G)=d+1$. Finally, we show that for any $d$-regular graph $G$ which does not contain 4-cycle and $\kappa(G)\leq\frac{d+1}{2}$, $\phi(G)=d+1$

Dynamic Chromatic Number of Regular Graphs

A dynamic coloring of a graph $G$ is a proper coloring such that for every vertex $v\in V(G)$ of degree at least 2, the neighbors of $v$ receive at least 2 colors. It was conjectured [B. Montgomery. {\em Dynamic coloring of graphs}. PhD thesis, West Virginia University, 2001.] that if $G$ is a $k$-regular graph, then $\chi_2(G)-\chi(G)\leq 2$. In this paper, we prove that if $G$ is a $k$-regular graph with $\chi(G)\geq 4$, then $\chi_2(G)\leq \chi(G)+\alpha(G^2)$. It confirms the conjecture for all regular graph $G$ with diameter at most 2 and $\chi(G)\geq 4$. In fact, it shows that $\chi_2(G)-\chi(G)\leq 1$ provided that $G$ has diameter at most 2 and $\chi(G)\geq 4$. Moreover, we show that for any $k$-regular graph $G$, $\chi_2(G)-\chi(G)\leq 6\ln k+2$. Also, we show that for any $n$ there exists a regular graph $G$ whose chromatic number is $n$ and $\chi_2(G)-\chi(G)\geq 1$. This result gives a negative answer to a conjecture of [A. Ahadi, S. Akbari, A. Dehghan, and M. Ghanbari. \newblock On the difference between chromatic number and dynamic chromatic number of graphs. \newblock {\em Discrete Math.}, In press].Comment: 8 page