On Murty-Simon Conjecture II

A graph is diameter two edge-critical if its diameter is two and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter two edge-critical graph on $n$ vertices is at most $\lfloor \frac{n^{2}}{4} \rfloor$ and the extremal graph is the complete bipartite graph $K_{\lfloor \frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil}$. In the series papers [7-9], the Murty-Simon Conjecture stated by Haynes et al. is not the original conjecture, indeed, it is only for the diameter two edge-critical graphs of even order. In this paper, we completely prove the Murty-Simon Conjecture for the graphs whose complements have vertex connectivity $\ell$, where $\ell = 1, 2, 3$; and for the graphs whose complements have an independent vertex cut of cardinality at least three.Comment: 9 pages, submitted for publication on May 10, 201

Total domination and the Caccetta–Häggkvist conjecture

AbstractA total dominating set in a digraph G is a subset W of its vertices such that every vertex of G has an immediate successor in W. The total domination number of G is the size of the smallest total dominating set. We consider several lower bounds on the total domination number and conjecture that these bounds are strictly larger than g(G)−1, where g(G) is the number of vertices of the smallest directed cycle contained in G. We prove that these new conjectures are equivalent to the Caccetta–Häggkvist conjecture which asserts that g(G)−1<nr in every digraph on n vertices with minimum outdegree at least r>0