A β-perfect graph is a simple graph G such that χ(G') = β(G') for every induced subgraph G' of G, where χ(G') is the chromatic number of G', and β(G') is defined as the maximum over all induced subgraphs H of G' of the minimum vertex degree in H plus 1 (i.e., δ(H)+1). The vertices of a β-perfect graph G can be coloured with χ(G) colours in polynomial time (greedily). The main purpose of this paper is to give necessary and sufficient conditions, in terms of forbidden induced subgraphs,...

A constrained colouring or, more specifically, an (α, β)-colouring of a hypergraph H, is an assignment of colours to its vertices such that no edge of H contains less than α or more than β vertices with different colours. This notion, introduced by Bujtás and Tuza, generalises both classical hypergraph colourings and more general Voloshin colourings of hypergraphs. In fact, for r-uniform hypergraphs, classical colourings correspond to (2, r)-colourings while an important instance of Voloshin colourings...

The $k$-core of a graph $G$, $C_k\left(G\right)$, is the maximal induced subgraph $H\subseteq G$ such that $\delta \left(G\right)\ge k$, if it exists. For $k>0$, the $k$-shell of a graph $G$ is the subgraph of $G$ induced by the edges contained in the $k$-core and not contained in the $(k+1)$-core. The core number of a vertex is the largest value for $k$ such that $v\in C_k\left(G\right)$, and the maximum core number of a graph, $\widehat{C}\left(G\right)$, is the maximum of the core numbers of the vertices of $G$. A graph $G$ is $k$-monocore if $\widehat{C}\left(G\right)=\delta \left(G\right)=k$. This paper discusses some basic results on the structure of $k$-cores and $k$-shells....

This paper gives a structure theorem for the class of countable 1-transitive coloured linear orderings for a countably infinite colour set, concluding the work begun in [1]. There we gave a complete classification of these orders for finite colour sets, of which there are ℵ₁. For infinite colour sets, the details are considerably more complicated, but many features from [1] occur here too, in more marked form, principally the use (now essential it seems) of coding trees, as a means of describing...

We prove that for any $c\ge 5$, there exists an infinite family ${\left(G_n\right)}_{n\in \mathbb{N}}$ of graphs such that $\chi \left(G_n\right)>c$ for all $n\in \mathbb{N}$ and $\chi (G_m\times G_n)\le c$ for all $m\ne n$. These counterexamples to Hedetniemi’s conjecture show that the Boolean lattice of exponential graphs with $K_c$ as a base is infinite for $c\ge 5$.

Let ₁,₂,...,ₙ be graph properties, a graph G is said to be uniquely (₁,₂, ...,ₙ)-partitionable if there is exactly one (unordered) partition V₁,V₂,...,Vₙ of V(G) such that $G\left[V_i\right]{\in }_i$ for i = 1,2,...,n. We prove that for additive and induced-hereditary properties uniquely (₁,₂,...,ₙ)-partitionable graphs exist if and only if ${}_i$ and ${}_j$ are either coprime or equal irreducible properties of graphs for every i ≠ j, i,j ∈ 1,2,...,n.

Snarks are bridgeless cubic graphs with chromatic index χ' = 4. A snark G is called critical if χ'(G-{v,w}) = 3, for any two adjacent vertices v and w. For any k ≥ 2 we construct cyclically 5-edge connected critical snarks G having an independent set I of at least k vertices such that χ'(G-I) = 4. For k = 2 this solves a problem of Nedela and Skoviera [6].