Let $G$ be a simple graph with maximum degree $\Delta\ge 3$, and let $P(G,k)$ denote its chromatic polynomial. For each positive integer $k$, the list-color function $P_{\ell}(G,k)$ is the minimum number of $L$-colorings of $G$ over all $k$-assignments $L$. In this paper, we prove that $P_{\ell}(G,k)=P(G,k)$ for every integer $k\ge 23.41\Delta$. This gives a threshold for equality that is linear in the maximum degree and independent of the number of vertices or edges. It improves the known sufficient condition $k\ge |E(G)|-1$ for graphs with sufficiently many edges relative to their maximum degree.
For an integer $k\geq2$, let $\chi_k'(G)$ denote the minimum number of colors in an edge-coloring of a graph $G$ such that every nonzero degree in each color subgraph is congruent to $1\pmod{k}$. A graph is a $0_k$-graph if every vertex degree is divisible by $k$. We disprove a conjecture of Berthe et al.\ (On modular...
For any graph $G$, let $P(G,k)$ and $P_{\ell}(G,k)$ denote the chromatic polynomial and the list-color function of $G$, respectively. It remains an open problem whether, for every graph $G$ and integer $k$, the equality $P(G,k)=P_{\ell}(G,k)>0$ implies that $P(G,k+1)=P_{\ell}(G,k+1)$ also holds. In this paper, we answe...
A B-coloring of a graph $G$ is a proper edge-coloring in which every $4$-cycle receives four distinct colors; let $q_B(G)$ be the minimum number of colors in such a coloring. Every graph of maximum degree $\Delta$ is $K_{2,\Delta+1}$-free; hence the known $2\Delta$ bound for planar graphs with $\Delta\ge38$ (Kong et al...
For a graph $G$, we write $\mathrm{mad}(G)$ for its maximum average degree and $\mathrm{diam} G$ for its diameter. Let $R_k(G)$ be the graph whose vertices are the proper colorings of $G$ with $k$ colors, where two colorings are adjacent when they differ at one vertex. Feghali (JCTB, 2021) proved that, for fixed intege...
For a graph $G$, a proper edge coloring of $G$ is called a D-coloring if every diamond subgraph of $G$ is rainbow. Let $\chi'_D(G)$ be the D-chromatic index of $G$, which is the smallest integer $k$ such that $G$ admits a D-coloring with $k$ colors. Let $\Delta$ be the maximum degree of $G$. The only known Brooks-type...
The adjacent vertex distinguishing (AVD)-total chromatic number $\chi''_{a}(G)$ of a graph $G$ is the least integer $k$ for which $G$ has a proper total coloring $f$ with $k$ colors such that $C_G(u)\neq C_G(v)$ for every edge $uv\in E(G)$, where $C_G(u)=\{f(u)\}\cup\{f(uw):uw\in E(G)\}$. The AVD-total coloring conject...
A. Banerjee, J. Geetha, K. Somasundaram· 0 citations
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