The Total Domination Number And Annihilation Number Of An Unicyclic Graph

Since 1960 s, as a young branch of mathematics, the graph theory has experienced the explosion growth. It has extensive application in physics, chemistry, biology, network theory, information science, computer science and other ?elds. As a sub ?eld in combinatorial mathematics, the graph theory has attracted much attention from all perspectives.For a nontrivial graph, there is a relationship between the total domination and annihilation number. The two parameters have many applications in various areas, for example,the annihilation number is a recent polynomial time computable upper bound for the independence number. We focus on the relationship of the two parameters of unicyclic graphs.For a simple graph G, γt(G) and a(G) denote the total domination number and annihilation number of G, respectively. Desormeaux, Haynes and Henning(Discrete Applied Mathematics. 161(2013) 349-354) pose a conjecture that if G is a connected nontrivial graph, then γt(G) ≤ a(G) + 1. They prove that this conjecture is true for trees. Our aim in this paper is to prove that γt(G) ≤ a(G) + 1 for an unicyclic graph G of order n with equality if and only if G～= Cn and n is not divisible by 4.