P.M.SUDHA, et. al. International Journal of Engineering Research and Applications www.ijera.com ISSN: 2248-9622, Vol. 11, Issue 10, (Series-I) October 2021, pp. 01-09 www.ijera.com DOI: 10.9790/9622-1110010109 1 | Page Cordial Decomposition in Various Graphs P.M.SUDHA 1* , P.SENTHILKUMAR 2 and S.VENGATAASALAM 3 1 Research Scholar, PG and Research Department of Mathematics, Government Arts and Science College, Kangeyam, Tiruppur – 638108, Tamil Nadu, India. 2 Assistant Professor, PG and Research Department of Mathematics, Government Arts and Science College, Kangeyam, Tiruppur – 638108, Tamil Nadu, India. 3 Professor, Department of Mathematics, Kongu Engineering College, Perundurai, Erode - 638060, Tamil Nadu, India. ABSTRACT: The main goal of this paper is to introduce and investigate the results of cordial decomposition and cordial decomposition number ) (G C  of a graphs. Also investigate some bounds of ) (G C  in product graphs like Cartesian product, composition etc. Keywords: Decomposition, labeling, cordial graphs, cordial decomposition and prime decomposition number. --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 20-09-2021 Date of Acceptance: 05-10-2021 --------------------------------------------------------------------------------------------------------------------------------------- I. INTRODUCTION In this Chapter, we define cordial decomposition and cordial decomposition number ) (G C  of a graphs. Also investigate some bounds of ) (G C  in product graphs like Cartesian product, composition etc. A bijection   1 , 0 ) ( :  G V f is called binary vertex labeling of ) , ( E V G and ) (v f is called the label of the vertex G v  under f . For an edge uv e  , the induced edge labeling   1 , 0 ) ( : *  G E f is given by ) ( ) ( ) ( * v f u f e f   . Let ) 1 ( ), 0 ( f f v v be the number of vertices of G having labels 0 and 1 respectively under f and ) 1 ( ), 0 ( f f e e be the number of edges having labels 0 and 1 respectively under * f . A graph ) , ( E V G is cordial if it admits cordial labelling. In this paper, we investigate the cordial decomposition for join and composition of some graphs. II. RESULTS ON CORDIAL DECOMPOSITION In this work, we investigate the cordial labeling for join and composition of some graphs. Definition 2.1: Let ) , ( E V G be a graph. A mapping   1 , 0 ) ( :  G V f is called binary vertex labeling of ) , ( E V G and ) (v f is called the label of the vertex G v  under f . For an edge uv e  , the induced edge labeling   1 , 0 ) ( : *  G E f is given by ) ( ) ( ) ( * v f u f e f   . Let ) 1 ( ), 0 ( f f v v be the number of vertices of G having labels 0 and 1 respectively under f and ) 1 ( ), 0 ( f f e e be the number of edges having labels 0 and 1 respectively under * f . A graph ) , ( E V G is cordial if it admits cordial labelling. Definition 2.2: A decomposition of G is a collection   r C H H H ,..... , 2 1   such that i H are edge disjoint and every edges in i H belongs to G . If each i H is a cordial graphs, then C  is called a cordial decomposition of G . The minimum cardinality of a cordial decomposition of G is called the cordial decomposition number of G and it is denoted by ). (G C  Theorem 2.1. The upper bounds of cordial decomposition number of the complete bipartite graph n m K , is ). ( ) ( , mn K n m C   Proof: The complete bipartite graphs n m K , having the set of vertices       n j v m i u V j i       1 1 . Note that RESEARCH ARTICLE OPEN ACCESS