A graph $G$ is $k$-choosable if it has a proper coloring for every $k$-list assignment. While every $C_3$-free planar graph is $4$-choosable, some of them are not $3$-choosable, as constructed by Voigt. Hu and Zhu conjectured that if $G$ is a $C_3$-free planar graph and $X \subseteq V(G)$ induces a bipartite subgraph, then $G$ has a proper $L$-coloring whenever $|L(x)| = 3$ for $x \in X$ and $|L(v)| = 4$ for $v \in V(G) \setminus X$. As evidence, they proved the conjecture when $X$ is an independent set. We provide further evidence by proving the conjecture when the induced subgraph $G[X]$ is an induced sparse matching. This is the first result supporting the conjecture in which the set $X$ receiving smaller lists may induce a subgraph with edges.
For a graph $G$, a proper coloring of $G$ is called proper conflict-free if for every non-isolated vertex $u$, there is at least one color appearing exactly once in $N_G(u)$. A graph $G$ is proper conflict-free $f$-choosable if for every list assignment $L$ with $|L(v)|\ge f(v)$ for each vertex $v$, $G$ admits a proper...
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...
A $k$-irregular graph is a graph with maximum degree $k$ such that vertices of degree $k$ are not adjacent. A $2$-distance $k$-coloring of a graph is a coloring of the vertices using $k$ colors in which any two vertices at distance at most $2$ receive distinct colors. The $2$-distance chromatic number of $G$, denoted b...
It is proved that PCF-COLORABILITY is NP-complete for bipartite graphs, and linear-time algorithms for PCF-COLORABILITY are provided in block graphs, proper interval graphs, chain graphs, and pseudo-split graphs.
For finite simple graphs $G,H$ on a common vertex set $V$, we say that $H$ is $G$-colorable if $H$ admits a proper list coloring with list assignment $L(v)=N_G(v)$ for all $v\in V$. This notion of coloring a graph using the neighborhood of another graph on the same vertex set, which we call \emph{graphic list coloring}...
A B-coloring of a graph $G$ is a proper edge-coloring in which every $4$-cycle is rainbow. Let $q_B(G)$ be the minimum number of colors in such a coloring. Gy\'arf\'as and S\'ark\"ozy (2023) determine $q_B(G)$ when $G=P_m\square P_n$ is a rectangular grid. In this paper, we completely determine $q_B(G)$ for cylindrical...
Zheng Jiang, Jia-Ao Li· 0 citations
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