A new lower bound for the total domination number in graphs proving a Graffiti.pc Conjecture

Abstract The total domination number γ t ( G ) of a graph G is the minimum cardinality of a set S of vertices so that every vertex of G is adjacent to a vertex in S . Our main result of this paper is that if G is a connected graph and x 1 , x 2 , x 3 ∈ V ( G ) , then γ t ( G ) ≥ 1 4 ( d ( x 1 , x 2 ) + d ( x 1 , x 3 ) + d ( x 2 , x 3 ) ) . Furthermore if equality holds in this bound, then the multiset { d ( x 1 , x 2 ) , d ( x 1 , x 3 ) , d ( x 2 , x 3 ) } is equal to { 2 , 3 , 3 } modulo four. As a consequence of this result, we prove a conjecture on the total domination number. To state this conjecture, let B denote the set of vertices of maximum eccentricity in G and let ecc ( B ) denote the maximum distance in G of a vertex outside B to a vertex of B . The following conjecture is known as Graffiti.pc Conjecture #233 ( http://cms.dt.uh.edu/faculty/delavinae/research/wowII ): if G is a connected graph of order at least two, then γ t ( G ) ≥ 2 ( ecc ( B ) + 1 ) / 3 . We prove this conjecture. In fact, as a consequence of our main result stated earlier we prove the following much stronger result: if G is a connected graph of order at least two, then γ t ( G ) ≥ ( 3 ecc ( B ) + 2 ) / 4 . We also prove that if G is a connected graph and x 1 , x 2 , x 3 , x 4 ∈ V ( G ) , then γ t ( G ) ≥ 1 8 ( d ( x 1 , x 2 ) + d ( x 1 , x 3 ) + d ( x 1 , x 4 ) + d ( x 2 , x 3 ) + d ( x 2 , x 4 ) + d ( x 3 , x 4 ) ) , and this result is best possible.