Roman domination in regular graphs

Abstract

A Roman domination function on a graph G = ( V ( G ) , E ( G ) ) is a function f : V ( G ) → { 0 , 1 , 2 } satisfying the condition that every vertex u for which f ( u ) = 0 is adjacent to at least one vertex v for which f ( v ) = 2 . The weight of a Roman dominating function is the value f ( V ( G ) ) = ∑ u ∈ V ( G ) f ( u ) . The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G . Cockayne et al. [E. J. Cockayne et al. Roman domination in graphs, Discrete Mathematics 278 (2004) 11–22] showed that γ ( G ) ≤ γ R ( G ) ≤ 2 γ ( G ) and defined a graph G to be Roman if γ R ( G ) = 2 γ ( G ) . In this article, the authors gave several classes of Roman graphs: P 3 k , P 3 k + 2 , C 3 k , C 3 k + 2 for k ≥ 1 , K m , n for min { m , n } ≠ 2 , and any graph G with γ ( G ) = 1 ; In this paper, we research on regular Roman graphs and prove that: (1) the circulant graphs C ( n ; { 1 , 3 } ) ( n ≥ 7 , n ⁄ ≡ 4 ( mod 5 ) ) and C ( n ; { 1 , 2 , … , k } ) ( k ≤ ⌊ n 2 ⌋ ) , n ⁄ ≡ 1 (mod ( 2 k + 1 ) ), ( n ≠ 2 k ) are Roman graphs, (2) the generalized Petersen graphs P ( n , 2 k + 1 ) ( n ≠ 4 k + 2 , n ≡ 0 (mod 4) and 0 ≤ k ≤ ⌊ n 2 ⌋ ), P ( n , 1 ) ( n ⁄ ≡ 2 (mod 4)), P ( n , 3 ) ( n ≥ 7 , n ⁄ ≡ 3 (mod 4)) and P ( 11 , 3 ) are Roman graphs, and (3) the Cartesian product graphs C 5 m □ C 5 n ( m ≥ 1 , n ≥ 1 ) are Roman graphs.