A Comparison on the Bounds of Chromatic Preserving Number and Dom-Chromatic Number of Cartesian Product and Kronecker Product of Paths
Let G be a simple graph with vertex set V and edge set E. A Set S Í V is said to be a chromatic preserving set or a cp-set if χ(<S>) = χ(G) and the minimum cardinality of a cp-set in G is called the chromatic preserving number or cp-number of G and is denoted by cpn(G). A cp-set of cardinality cpn(G) is called a cpn-set. A subset S of V is said to be a dom- chromatic set (or a dc-set) if S is a dominating set and χ(<S>) = χ(G). The minimum cardinality of a dom-chromatic set in a graph G is called the dom-chromatic number (or dc- number) of G and is denoted by γch(G). The Kronecker product G1 Ù G2 of two graphs G1 = (V1, E1) and G2 = (V2, E2) is the graph G with vertex set V1 x V2 and any two distinct vertices (u1, v1) and (u2, v2) of G are adjacent if u1u2Î E1 and v1v2Î E2. The Cartesian product G1 x G2 is the graph with vertex set V1 x V2 where any two distinct vertices (u1, v1) and (u2, v2) are adjacent whenever (i) u1 = u2 and v1v2 Î E2 or (ii) u1u2Î E1 and v1 = v2. These two products have no common edges. They are almost like complements but not exactly. In this paper, we discuss the behavior of the cp-number and dc-number and their bounds for product of paths in the two cases. A detailed comparative study is also done.
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