An L (2, 1)-coloring of a simple connected graph G is an assignment of non-negative integers to the vertices of G such that adjacent vertices color difference is at least two, and vertices that are at distance two from each other get different colors. The maximum color assigned in an L (2, 1)-coloring is called span of that coloring. The span of a graph G denoted by λ (G) is the smallest span taken over all L (2, 1)-colorings of G. A hole is an unused color within the range of colors used by the coloring. An L (2, 1)-coloring f is said to be irreducible if no other L (2, 1)-coloring can be produced by decreasing a color of f . The maximum number of holes of a graph G, denoted by H λ (G), is the maximum number of holes taken over all irreducible L (2, 1)-colorings with span λ (G). Laskar and Eyabi (Christpher, 2009) conjectured that if T is a tree, then H λ (T ) = 2 if and only if T = P n , n > 4. We show that this conjecture does not hold by providing a counterexample. Also, we give some classes of trees with maximum number of holes two.

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For any positive integer k, a k-distance coloring of a graph G is a vertex coloring of G in which no two vertices at distance less than or equal to k receive the same color. The k-distance chromatic number of G, denoted by χ k (G) is the smallest integer α for which G has a k-distance α-coloring. In this paper, we improve the lower bound for the k-distance chromatic number of an arbitrary graph for k odd case and see that trees achieve this lower bound by determining the k-distance chromatic number of trees. Also, we find k-distance chromatic number of cycles and 2-distance chromatic number of a graph G in which every pair of cycles are edge disjoint. c

A radio coloring of graphs is a modification of the frequency assignment problem. For a connected simple graph G, a mapping g of the vertices of G to the positive integers (colors) such that for every pair u and v of G, | g(u) − g(v)| is at least 1 + diam(G) − d(u, v), is called a radio coloring of G. The largest color used by g is called span of g, denoted by rn(g). The radio number, rn(G), is the least of { rn(g) : g is a radio coloring of G }. In this paper, for n ⩾ 7 we obtain the radio number of Cartesian product of complete graph K n and cycle C m , K n ☐C m , for n even and m odd, and for n odd and m 5 (mod 8).

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