Total and Entire Labeling of Graphs

Abstract

\A plane graph is a particular drawing of a planar graph on the Euclidean plane. Let G(V;E; F) be a plane graph with vertex set V , edge set E and face set F. A proper entire t-colouring of a plane graph is a mapping : V (G) [ E(G) [ F(G) 􀀀! f1; 2; : : : ; tg such that any two adjacent or incident elements in the set V (G)[E(G)[F(G) receive distinct colours. The entire chromatic number, denoted by vef (G), of a plane graph G is the smallest integer t such that G has a proper entire t-colouring. The proper entire t-colouring of a plane graph have been studied extensively in the literature. There are several modi cation on entire t-colouring. We focus on a face irregular entire k-labeling of a 2-connected plane graph as a labeling of vertices, edges and faces of G with labels from the set f1; 2; : : : ; kg in such a way that for any two di erent faces their weights are distinct. The weight of a face under a k-labeling is the sum of labels carried by that face and all the edges and vertices incident with the face. The minimum k for which a plane graph G has a face irregular entire k-labeling is called the entire face irregularity strength". \Another variation to entire t-colouring is a d-antimagic labeling as entire labeling of a plane graph with the property that for every positive integer s, the weights of s-sided faces form an arithmetic sequence with a common di erence d. In the thesis, we estimate the bounds of the entire face irregularity strength for disjoint union of multiple copies of a plane graph and prove the sharpness of the lower bound in two cases. Also we study the existence of d-antimagic labelings for vi the Klein-bottle fullerene that is for a nite trivalent graph embedded on the Kleinbottle with each face is a hexagon. In last chapter we investigate the 3-total edge product cordial labeling of hexagonal grid (honeycomb) that is the planar graph with m rows and n columns of hexagons".