Majority C-coloring of graphs

Inspired by the majority colorings and C-colorings, we introduce and study the majority C-coloring of graphs. In such a vertex coloring, every vertex shares its color with at least half of its neighbors. The maximum number of colors that can be used in a majority C-coloring of a graph $G$ is called the majority C-chromatic number and denoted by $\mc(G)$. An upper bound on $\mc(G)$ is proved in terms of the order, minimum, and maximum degree. Its sharpness is demonstrated by several results over different graph classes. In particular, $\mc(P_n^k)= \mc(C_n^k)= \lfloor n/(k+1)\rfloor$ is true for the $k$-th power of a path and a cycle if $n \ge k+1$. Further, $\mc(G) = (n-d)/3$ holds if $G$ is a $(\mbox{claw}, K_4)$-free cubic graph and contains $d$ diamonds. %claw-free cubic graph on $n \ge 6$ vertices and contains $d$ diamonds. It is further shown that the majority C-chromatic number is not monotone under edge deletion. In fact, both the lower and upper bounds are sharp in the inequality chain $\mc(G)-2 \leq \mc(G-e) \leq \mc(G) +1$. The minimum and maximum number of edges in an $n$-vertex graph $G$ with $\mc(G)=k$ are determined for every $n$ and $k$. It is also pointed out that the classical chromatic number $\chi(G)$ and $\mc(G)$ are incomparable, and the difference $\mc(G)-\chi(G)$ can take any positive or negative integer. On the other hand, $\mc(G)+\chi(G) \leq n+1$ holds for every graph $G$ of order $n$. The decision problem of whether $\mc(G) \ge k$ holds is NP-complete for every fixed $k\ge 2$. In contrast, some sufficient conditions for $\mc(G) \ge 2$ are proved, and a linear-time algorithm is presented that determines $\mc(T)$ if $T$ is a tree.