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Let Gbe a connected graph with V(G)={v1,…,vn}. The rainbow connection number rc(G)is the smallest kfor which there is a map γ:E(G)→{1,…,k}such that any two vertices can be connected by a path whose edge colors are all distinct. The generalized composition G[H1,…,Hn]is obtained by replacing each viwith the graph Hi. We prove rc(G[H1,…,Hn])=diam(G)if each Hihas at least diam(G)≥4vertices, improving known upper bounds of Basavaraju et al. and Gologranc et al. (2014). We prove the same result when diam(G)=3but with some additional conditions. When diam(G)=2, we show that the largest possible value of rc(G[H1,…,Hn])is related to whether every vertex of Gis contained in a triangle or not.

AbstractIn this note we present a few open problems on various aspects of graph labelings, which have not been included in any of the other papers appearing in this volume.

AbstractCharacterization of DNA graph gives important contribution in completing the computational step of Sequencing by Hybridization (SBH). Some graphs are already characterized as DNA graph using (α, k)-labeling. Dicycle and dipath are DNA graphs, while rooted trees and self adjoint digraphs are DNA graphs if and only if their maximum degree is not greater than four. In this paper we also use (α, k)-labeling to characterize line digraph of dicycle with one chord Ctnas DNA graph and show that for m, n∊ ℕ, are DNA graphs.

AbstractA total k-labeling is a map that carries vertices and edges of a graph Ginto a set of positive integer labels {1, 2,…, k}. An edge irregular total k-labeling of a graph Gis a total k-labeling such that the weights calculated for all edges are distinct. The weight of an edge uvin Gis defined as the sum of the label of u, the label of vand the label of uv. The total edge irregularity strength of G, denoted by tes(G), is the minimum value of the largest label kover all such edge irregular total k-labelings. In this paper, we investigate the total edge irregularity strength of generalized helm, Hmnfor n≥ 3, m= 1, 2, and m≡ 0 (mod3).

AbstractA graph G= (V, E) with |E| = qis said to be odd harmonious if there exists an injection f: V(G) → {0, 1, 2,…, 2q- 1} such that the induced function f*: E(G) → {1, 3, 5,…, 2q- 1} defined by f*(xy) = f(x)+f(y) is a bijection. Then fis said to be odd harmonious labeling of G. A dumbbell graph Dn,k,2is a bicyclic graph consisting of two vertex-disjoint cycles Cn, Ckand a path P2joining one vertex of Cnwith one vertex of Ck. A prism graph Cnx Pmis a Cartesian product of cycle Cnand path Pm. In this paper we show that the dumbbell graph Dn,k,2is odd harmonious for n≡ k≡ 0 (mod 4) and n≡ k≡ 2 (mod 4), generalized prism graph Cnx Pmis odd harmonious for n≡ 0 (mod 4) and for any m, and generalized prism graph Cnx Pmis not odd harmonious for n≡ 2 (mod 4).