More on the total dominator chromatic number of a graph

Abstract

Let G be a simple graph. A total dominator coloring of G, is a proper coloring of the vertices of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number χ t d (G) of G, is the minimum number of colors among all total dominator coloring of G. The neighbourhood corona of two graphs G 1 and G 2 is denoted by G 1 ⋆ G 2 and is the graph obtained by taking one copy of G 1 and |V (G 1 )| copies of G 2 , and joining the neighbours of the ith vertex of G 1 to every vertex in the ith copy of G 2 . In this paper, we study the total dominator chromatic number of the neighbourhood of two graphs and investigate the total dominator chromatic number of r-gluing of two graphs. Stability (bondage number) of total dominator chromatic number of G is the minimum number of vertices (edges) of G whose removal changes the TDC-number of G. We study the stability and bondage number of certatin graphs.

On strict strong coloring of central graphs

“…The minimum number of colors required for a strict strong coloring of G is called the strict strong chromatic number of G and is denoted by χ ss (G). Some basic results on strict strong colorings are given in [2,10,13,17]. Given a graph G, we subdivide each edge of G exactly once and join all the non-adjacent vertices of G. The graph obtained by this process is called central graph of G denoted by C(G).…”

The Dominator Edge Coloring of Graphs

“…A concept closely related to dominator edge coloring is total dominator edge coloring. In [12], Ghanbari and Alikhani introduced the total dominator edge coloring. A total dominator edge coloring of G, briefly TDE-coloring, is a proper edge coloring in which each edge of G is adjacent to every edge of some (other) color class.…”

Introduction to dominated edge chromatic number of a graph

“…For a graph G with no isolated vertex, the total dominator coloring is a proper coloring of G in which each vertex of the graph is adjacent to every vertex of some (other) color class. The total dominator chromatic number, abbreviated TDchromatic number, χ t d (G) of G is the minimum number of color classes in a TD-coloring of G. For more information see [5,6]. A set T of vertices is a total dominating set of G if every vertex of G is adjacent to at least one vertex in T .…”

“…G 1 = G, and if the graph G has v vertices and e edges, then the graph G has v + (k − 1)e vertices and ke edges. In this section we study dominated edge coloring of k-subdivision of a graph ([7]). In Particular, we obtain some bounds for χ ′ dom () and prove that for any…”