Edge Roman Domination on Graphs

An edge Roman dominating function of a graph G is a function f : E ( G ) → { 0 , 1 , 2 } satisfying the condition that every edge e with f ( e ) = 0 is adjacent to some edge e ′ with f ( e ′ ) = 2 . The edge Roman domination number of G , denoted by γ R ′ ( G ) , is the minimum weight w ( f ) = ∑ e ∈ E ( G ) f ( e ) of an edge Roman dominating function f of G . This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad stating that if G is a graph of maximum degree Δ on n vertices, then γ R ′ ( G ) ≤ ⌈ Δ Δ + 1 n ⌉ . While the counterexamples having the edge Roman domination numbers 2 Δ - 2 2 Δ - 1 n , we prove that 2 Δ - 2 2 Δ - 1 n + 2 2 Δ - 1 is an upper bound for connected graphs. Furthermore, we provide an upper bound for the edge Roman domination number of k -degenerate graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic graphs. In addition, we prove that the edge Roman domination numbers of planar graphs on n vertices is at most 6 7 n , which confirms a conjecture of Akbari and Qajar. We also show an upper bound for graphs of girth at least five that is 2-cell embeddable in surfaces of small genus. Finally, we prove an upper bound for graphs that do not contain K 2 , 3 as a subdivision, which generalizes a result of Akbari and Qajar on outerplanar graphs.