In this work we consider arc criticality in colourings of oriented graphs. We study deeply critical graphs, those graphs for which the removal any results a decrease chromatic number by 2. prove existence cliques every odd order \(n\ge 9\), closing an open question posed Borodin et al. [Journal Combinatorial Theory, Series B, 81(1):150–155, 2001]. Additionally, non-existence among family circul...

A set S ⊆ V (G) of an undirected graph G is a clique if every two distinct vertices in are adjacent. superclique for pair v, w ∈ S, there exists u \ such that NG(v) NG(w) or NG(v). The maximum cardinality (resp. superclique) called the number G. In this paper, we determine and numbers some graphs.

Recently Cutler and Radcliffe proved that the graph on $n$ vertices with maximum degree at most $r$ having cliques is a disjoint union of $\lfloor n/(r+1)\rfloor$ size $r+1$ together clique remainder vertices. It very natural also to consider this question when limiting resource edges rather than In paper we prove among graphs $m$ $r$, has least two $\bigl\lfloor m \bigm/\binom{r+1}{2} \bigr\rf...

It is well known that a clique with k + 1 vertices is the only minimal obstruction to k-colourability of chordal graphs. A similar result is known for the existence of a cover by cliques. Both of these problems are in fact partition problems, restricted to chordal graphs. The first seeks partitions into k independent sets, and the second is equivalent to finding partitions into cliques. In an e...

A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color is equal to the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. Coordinated graphs are a subclass of perfect graphs. In a previous work, polynomial time algorithms were found...

The Turán connected graph TCn,α is obtained from α cliques of size b αc or d αe by joining all cliques by an edge to one central vertex in one of the larger cliques. The graph TCn,α was shown recently by Bruyère and Mélot to maximise the number of independent sets among connected graphs of order n and independence number α. We prove a generalisation of this result by showing that TCn,α in fact ...

A Turán connected graph TCn,α is obtained from α cliques of size b αc or d αe by joining all cliques by an edge to one central vertex in one of the larger cliques. The graph TCn,α was shown recently by Bruyère and Mélot to maximise the number of independent sets among connected graphs of order n and independence number α. We prove a generalisation of this result by showing that TCn,α in fact ma...

Potential maximal cliques and minimal separators are combinatorial objects which were introduced and studied in the realm of minimal triangulation problems including Minimum Fill-in and Treewidth. We discover unexpected applications of these notions to the field of moderate exponential algorithms. In particular, we show that given an n-vertex graph G together with its set of potential maximal c...