A NOTE ON GENERALIZED RAINBOW CONNECTION OF CONNECTED GRAPHS AND THEIR NUMBER OF EDGES

Let l ≥ 1, k ≥ 1 be two integers. Given an edge-coloured connected graph G. A path P in the graph G is called l-rainbow path if each subpath of length at most l + 1 is rainbow. The graph G is called (k, l)-rainbow connected if any two vertices in G are connected by at least k pairwise internally vertex-disjoint l-rainbow paths. The smallest number of colours needed in order to make G (k, l)-rainbow connected is called the (k, l)-rainbow connection number of G and denoted by rck,l(G). In this paper, we first focus to improve the upper bound of the (1, l)-rainbow connection number depending on the size of connected graphs. Using this result, we characterize all connected graphs having the large (1, 2)-rainbow connection number. Moreover, we also determine the (1, l)-rainbow connection number in a connected graph G containing a sequence of cut-edges.

The Rainbow Connection Number of Triangular Snake Graphs

Rainbow coloring problems, of noteworthy applications in Information Security, have been receiving much attention last years in Combinatorics. The rainbow connection number of a graph G is the least number of colors for a (not necessarily proper) edge coloring of G such that between any pair of vertices there is a path whose edge colors are all distinct. In this paper we determine the rainbow connection number of the triple triangular snake graphs.

Rainbow connection number of Cm o Pn and Cm o Cn

Let <em>G </em>= (<em>V</em>(<em>G</em>),<em>E</em>(<em>G</em>)) be a nontrivial connected graph. A rainbow path is a path which is each edge colored with different color. A rainbow coloring is a coloring which any two vertices should be joined by at least one rainbow path. For two different vertices, <em>u,v</em> in <em>G</em>, a geodesic path of <em>u-v</em> is the shortest rainbow path of <em>u-v</em>. A strong rainbow coloring is a coloring which any two vertices joined by at least one rainbow geodesic. A rainbow connection number of a graph, denoted by <em>rc</em>(<em>G</em>), is the smallest number of color required for graph <em>G</em> to be said as rainbow connected. The strong rainbow color number, denoted by <em>src</em>(<em>G</em>), is the least number of color which is needed to color every geodesic path in the graph <em>G</em> to be rainbow. In this paper, we will determine the rainbow connection and strong rainbow connection for Corona Graph <em>Cm</em> o <em>Pn</em>, and <em>Cm</em> o <em>Cn</em>.