A sharp upper bound on the average distance of a graph depending on the order and the independence number is given. As a corollary we obtain the maximum average distance of a graph with given order and matching number. All extremal graphs are determined.

In 1965, Vizing conjectured that the independence ratio of edge-chromatic critical graphs is at most 1 2 . We prove that for every > 0 this conjecture is equivalent to its restriction on a specific set of edge-chromatic critical graphs with independence ratio smaller than 1 2 + .

let $g$ be a finite group with the identity $e$‎. ‎the subgroup intersection graph $gamma_{si}(g)$ of $g$ is the graph with vertex set $v(gamma_{si}(g)) = g-e$ and two distinct vertices $x$ and $y$ are adjacent in $gamma_{si}(g)$ if and only if $|leftlangle xrightrangle capleftlangle yrightrangle|>1$‎, ‎where $leftlangle xrightrangle $ is the cyclic subgroup of $g$ generated by $xin g$‎. ‎in th...

A 2-hued coloring of a graph G (also known as conditional (k, 2)-coloring and dynamic coloring) is a coloring such that for every vertex v ∈ V (G) of degree at least 2, the neighbors of v receive at least 2 colors. The smallest integer k such that G has a 2-hued coloring with k colors, is called the 2-hued chromatic number of G and denoted by χ2(G). In this paper, we will show that if G is a re...

An independent set Ic is a critical independent set if |Ic|−|N(Ic)| ≥ |J | − |N(J)|, for any independent set J . The critical independence number of a graph is the cardinality of a maximum critical independent set. This number is a lower bound for the independence number and can be computed in polynomial-time. Any graph can be decomposed into two subgraphs where the independence number of one s...

We prove that every digraph of independence number at most 2 and arc-connectivity least has an out-branching B + in-branching − which are arc-disjoint (we call such branchings a good pair). This is best possible in terms the as there infinitely many strong digraphs with arbitrarily high minimum in- out-degrees have no pair. The result settles conjecture by Thomassen for 2. on 6 vertices pair gi...