On Locally Identifying Coloring of Cartesian Product and Tensor Product of Graphs
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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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