Lower and upper orientable strong diameters of graphs satisfying the Ore condition

Abstract

Let D be a strong digraph. The strong distance between two vertices u and v in D , denoted by s d D ( u , v ) , is the minimum size (the number of arcs) of a strong sub-digraph of D containing u and v . For a vertex v of D , the strong eccentricity s e ( v ) is the strong distance between v and a vertex farthest from v . The minimum strong eccentricity among all vertices of D is the strong radius, denoted by s r a d ( D ) , and the maximum strong eccentricity is the strong diameter, denoted by s d i a m ( D ) . The lower (resp. upper) orientable strong radius s r a d ( G ) (resp. S R A D ( G ) ) of a graph G is the minimum (resp. maximum) strong radius over all strong orientations of G . The lower (resp. upper) orientable strong diameter s d i a m ( G ) (resp. S D I A M ( G ) ) of a graph G is the minimum (resp. maximum) strong diameter over all strong orientations of G . In this work, we determine a bound of the lower orientable strong diameters and the bounds of the upper orientable strong diameters for graphs G = ( V , E ) satisfying the Ore condition (that is, σ 2 ( G ) = min { d ( x ) + d ( y ) | ∀ x y ∉ E ( G ) } ≥ n ), in terms of girth g and order n of G .