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Abstract. In an undirected graph, a proper (k, i)-coloring is an assignment of a set of k colors to each vertex such that any two adjacent vertices have at most i common colors. The (k, i)-coloring problem is to compute the minimum number of colors required for a proper (k, i)-coloring. This is a generalization of the classic graph coloring problem. Majumdar et. al. [CALDAM 2017] studied this problem and showed that the decision version of the (k, i)-coloring problem is fixed parameter tractable (FPT) with tree-width as the parameter. They asked if there exists an FPT algorithm with the size of the feedback vertex set (FVS) as the parameter without using tree-width machinery. We answer this in positive by giving a parameterized algorithm with the size of the FVS as the parameter. We also give a faster and simpler exact algorithm for (k, k − 1)-coloring, and make progress on the NP-completeness of specific cases of (k, i)-coloring.

The closed neighborhood conflict-free chromatic number of a graph G, denoted by χ G ( )

For a positive integer k, a proper k-coloring of a graph G is a mapping f : V (G) → {1, 2, . . . , k} such that f (u) ̸ = f (v) for each edge uv ∈ E(G). The smallest integer k for which there is a proper k-coloring of G is called chromatic number of G, denoted by χ(G). A locally identifying coloring (for short, lid-coloring) of a graph G is a proper k-coloring of G such that every pair of adjacent vertices with distinct closed neighborhoods has distinct set of colors in their closed neighborhoods. The smallest integer k such that G has a lid-coloring with k colors is called locally identifying chromatic number (for short, lid-chromatic number ) of G, denoted by χ lid (G). In this paper, we study lid-coloring of Cartesian product and tensor product of two graphs. We prove that if G and H are two connected graphs having at least two vertices then (a) χ lid (G□H) ≤ χ(G)χ(H) − 1 and (b) χ lid (G × H) ≤ χ(G)χ(H). Here G□H and G × H denote the Cartesian and tensor products of G and H respectively. We also give exact values of lid-chromatic number of Cartesian product (resp. tensor product) of two paths, a cycle and a path, and two cycles.

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