Colorful Graph Coloring

Given a simple graph G and a positive integer d, the Colorful Graph Coloring problem (CGC) asks for the minimum number of colors needed to color the “coloring elements” of G, such that for every “colorful element” of G, the coloring elements in its “neighborhood” have at least d colors. Both coloring elements and colorful elements of G can be vertices and edges, which means that there are four variants of CGC. With both coloring and colorful elements being vertices, we use Vertex-Coloring Vertex-Colorful (VCVC) to denote the variant of CGC, which asks for the minimum number of colors needed to color the vertices such that for every vertex v, the vertices in the closed neighborhood of v are colored by at least d colors. The Vertex-Coloring Edge-Colorful variant (VCEC) colors the vertices such that every edge is incident to vertices of at least d colors. Clearly, this variant is meaningful only for  $$d\le 2$$ with simple graphs as input, and with  $$d=2$$ is equivalent to the classical Graph Coloring problem. The Edge-Coloring Vertex-Colorful variant (ECVC) demands for every vertex v that the edges incident to v are colored by at least d colors. Finally, the Edge-Coloring Edge-Colorful variant (ECEC) colors the edges such that the “closed neighborhood” of every edge contains d distinctly colored edges. The closed neighborhood of an edge e contains e and all edges sharing endpoints with e. Motivated by the extensive research on Graph Coloring and the applications of Colorful Graph Coloring in resource allocation, we initialize the complexity study of VCVC, ECVC, and ECEC and achieve the following results. VCVC is polynomial-time solvable for  $$d\le 2$$ and becomes NP-hard for  $$d\ge 3$$ . We also present a parameterized algorithm for VCVC with treewidth as parameter. ECVC is NP-hard only in the case that d is set equal to the minimum degree of vertices  $$\delta (G)$$ . If  $$d\ne \delta (G)$$ , then ECVC is polynomial-time solvable. Based on this, we show that in the case of  $$d=\delta (G)$$ , we can compute a coloring with  $$d+1$$ colors in polynomial time, providing an absolute approximation with an additive term one. Moreover, we prove that ECEC is NP-hard with  $$d\ge 4$$ , and solvable in polynomial time with  $$d\le 3$$ . Finally, we present integer linear programming formulations for the problems.