Tight Upper Bound of the 3-Total-Rainbow Index for 2-(Edge-)Connected Graphs

Let [Formula: see text] be an edge-colored graph with order [Formula: see text] and [Formula: see text] be a fixed integer satisfying [Formula: see text]. For a vertex set [Formula: see text] of at least two vertices, a tree containing the vertices of [Formula: see text] in [Formula: see text] is called an [Formula: see text]-tree. The [Formula: see text]-tree [Formula: see text] is a total-rainbow [Formula: see text]-tree if the elements of [Formula: see text], except for the vertex set [Formula: see text], have distinct colors. A total-colored graph [Formula: see text] is said to be total-rainbow [Formula: see text]-tree connected if for every set [Formula: see text] of [Formula: see text] vertices in [Formula: see text], there exists a total-rainbow [Formula: see text]-tree in [Formula: see text], while the total-coloring of [Formula: see text] is called a [Formula: see text]-total-rainbow coloring. The [Formula: see text]-total-rainbow index of a nontrivial connected graph [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed in a [Formula: see text]-total-rainbow coloring of [Formula: see text]. In this paper, we show a sharp upper bound for [Formula: see text], where [Formula: see text] is a 2-connected or 2-edge-connected graph.

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>.

BILANGAN TERHUBUNG TITIK PELANGI PADA AMALGAMASI GRAF BERLIAN

Suppose there is a simple, and finite graph G = (V, E). The coloring of vertices c is denoted by c: E(G) → {1,2, ..., k} with k is the number of rainbow colors on graph G. A graph is said to be rainbow connected if every pair of points x and y has a rainbow path. A path is said to be a rainbow if there are not two edges that have the same color in one path. The rainbow connected number of graph G denoted by rc(G) is the smallest positive integer-k which makes graph G has rainbow coloring. Furthermore, a graph is said to be connected to rainbow vertex if at each pair of vertices x and y there are not two vertices that have the same color in one path. The rainbow vertex connected to the number of graph G is denoted by rvc(G) is the smallest positive integer-k which makes graph G has rainbow coloring. This paper discusses rainbow vertex-connected numbers in the amalgamation of a diamond graph. A diamond graph with 2n points is denoted by an amalgamation of a diamond graph by adding the multiplication of the graph t at point v is denoted by Amal (Brn,v,t).