Vertex Distinguishing Proper Edge Colorings As Well As Total Colorings Of Several Classes Of Graphs

Let G=(V, E) be a simple, undirected and fnite graph.A proper k-edge coloring f of a graph G is an assignment of k colors,1,2,···, k,to edges of G such that no two adjacent edges receive the same color. Given such acoloring f, for any vertex u∈V (G), let S(u) be the set of colors assigned to the edgesincident to u, if S(u)=S(v) for any two distinct vertices u and v of V (G), then we saythat f is a vertex-distinguishing proper edge coloring of graph G (in brief k VDPEC).The minimum number of colors required for a VDPEC of G, denoted by χ′s(G), iscalled the vertex-distinguishing proper edge chromatic number. In Chapter2, thevertex-distinguishing proper edge chromatic numbers of Kp[Pq], Kp[Sq], Kp[Wq].A proper k-total coloring f of a graph G is an assignment of k colors,1,2,···, k,to the vertices and edges of G such that no two adjacent vertices receive the samecolor, no two adjacent edges receive the same color, and no edge receives the samecolor as one of its endpoints. Given such a coloring f, for any vertex u∈V (G),let C(u) be the set of colors assigned to vertex u and edges incident to vertex u, ifC(u)=C(v), then we say that we say that f is a vertex-distinguishing proper totalcoloring of graph G (in brief k VDTC). The minimum number of colors requiredfor a VDTC of G, denoted by χvt(G), is called the vertex-distinguishing proper totalchromatic number. In Chapter3, the vertex-distinguishing proper total chromaticnumber of K(p×q) is obtained.