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A r-hued k-coloring of G is a proper coloring with k colors such that for every vertex v with degree d(v) in G, the color number of the neighbor of v is at least minfd(v); rg. The smallest integer k such that G has a r-hued k-coloring is called the r-hued chromatic number and denoted by r(G). In this paper, we study the r-hued coloring of Cartesian products of square of cycles with paths. [ABSTRACT FROM AUTHOR]

For a graph G and a positive integer k, a subset S of vertices of G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. The cardinality of a minimum k-path vertex cover is denoted by ψk(G). In this paper, we present the exact values of ψk in some product graphs of stars and complete graphs. [ABSTRACT FROM AUTHOR]

For a connected graph G = (V (G),E(G)), a vertex set S ⊆ V (G) is a k-restricted vertex-cut if G - S is disconnected such that every componen- t of G - S has at least k vertices. The k-restricted connectivity κk(G) of the graph G is the cardinality of a minimum k-restricted vertex-cut of G. In this paper, we give the 3-restricted connectivity and the 4-restricted connectivity of the Cartesian product graphs, and we proposed two conjectures for general cases of the k-restricted connectivity of the Cartesian product graphs. [ABSTRACT FROM AUTHOR]