Upper Bounds On The Signed Edge Domination Number Of A Graph

A signed edge domination function (or SEDF) of a simple graph G = (V, E) is a function f : E -> {1,-1} such that Sigma(e 'is an element of N[e]) f (e ') >= 1 holds for each edge e is an element of E, where N[e] is the set of edges in G that share at least one endpoint with e. Let gamma '(s)(G) denote the minimum value of f (G) among all SEDFs f, where f (G) = Sigma(e is an element of E)f (e). In 2005, Xu conjectured that gamma '(s)(G) <= n - 1, where n is the order of G. This conjecture has been proved for the two cases v(odd)(G) = 0 and v(even)(G) = 0, where v(odd)(G) (resp. v(even)(G)) is the number of odd (resp. even) vertices in G. This article proves Xu's conjecture for v(even)(G) is an element of{1, 2}. We also show that for any simple graph G of order n, gamma '(s)(G) <= n + v(odd)(G)/2 and gamma '(s)(G) <= n - 2 + v(even)(G) when v(even)(G) > 0, and thus gamma '(s)(G) <= (4n - 2)/3. Our result improves the best current upper bound of gamma '(s)(G) <= [3n/2]. (C) 2020 Elsevier B.V. All rights reserved.