Independence number of edge‐chromatic critical graphs

On the independence number of edge chromatic critical graphs

In 1968, Vizing conjectured that for any edge chromatic critical graph G = (V,E) with maximum degree ∆ and independence number α(G), α(G) ≤ |V | 2 . It is known that α(G) < 3∆−2 5∆−2 |V |. In this paper we improve this bound when ∆ ≥ 4. Our precise result depends on the number n2 of 2-vertices in G, but in particular we prove that α(G) ≤ 3∆−3 5∆−3 |V | when ∆ ≥ 5 and n2 ≤ 2(∆− 1).

Edge number critical triangle free graphs with low independence numbers

The structure of all triangle free graphs G = (V,E) with |E|−6|V |+α(G) = 0 is determined, yielding an affirmative answer to a question of Stanis law Radziszowsky and Donald Kreher.

K-independence Critical Graphs

Let k be a positive integer and G = (V (G), E(G)) a graph. A subset S of V (G) is a k-independent set of G if the subgraph induced by the vertices of S has maximum degree at most k − 1. The maximum cardinality of a k-independent set of G is the k-independence number βk(G). In this paper, we study the properties of graphs for which the k-independence number changes whenever an edge or vertex is ...

Semi-regular graphs of minimum independence number

There are many functions of the degree sequence of a graph which give lower bounds on the independence number of the graph. In particular, for every graph G, α(G) ≥ R(d(G)), where R is the residue of the degree sequence of G. We consider the precision of this estimate when it is applied to semi-regular degree sequences. We show that the residue nearly always gives the best possible estimate on ...

On the independence number of random graphs