Game Chromatic Number of Generalized Petersen Graphs and Jahangir Graphs

Ramy Shaheen,Khaled Alshehada,Ziad Kanaya,Kannan Krithivasan

doi:10.1155/2020/6475427

Abstract

Let G = V , E be a graph, and two players Alice and Bob alternate turns coloring the vertices of the graph G a proper coloring where no two adjacent vertices are signed with the same color. Alice's goal is to color the set of vertices using the minimum number of colors, which is called game chromatic number and is denoted by χ g G , while Bob's goal is to prevent Alice's goal. In this paper, we investigate the game chromatic number χ g G of Generalized Petersen Graphs G P n , k for k ≥ 3 and arbitrary n , n -Crossed Prism Graph, and Jahangir Graph J n , m .

Highlights

In [8], Shaheen et al investigated the game chromatic number for some special circulant graphs and Generalized Petersen Graphs when k 1, 2, 3
E Game Coloring Number colg(G) is the competitive version of the coloring number, which was introduced for the first time by Zhu in [10] to give an improved upper bound of the game chromatic number of planar graphs. e game coloring number is a two-person game, which is played by Alice and Bob, who alternate turns and with Alice starting first
A vertex is selected and is added to the end of the linear ordering which was formed by previous moves

Summary

Introduction

In [8], Shaheen et al investigated the game chromatic number for some special circulant graphs and Generalized Petersen Graphs when k 1, 2, 3. E Coloring Number was defined in [9] as the following: Let Π(G) be the set of linear orderings on the vertex set of a graph G, and let L ∈ Π(G). Let the integer k, n satisfy k < n/2; Generalized Petersen Graph GP(n, k) is the graph whose vertex set is V∪U where V 􏼈v1, ..., vn􏼉 and U 􏼈u1, ..., un􏼉 and its edge set is E 􏼈vivi+1, viui, uiui+k􏼉, where i 1, 2, ..., n and subscripts are reduced modulo n.

Results

Conclusion