A graph is k-vertex-critical if χ (G)= k but χ (Gv)< k for all v∈ V (G). We construct new infinite
families of k-vertex-critical (P 5, C 5)-free graphs for all k≥ 6. Our construction generalises …

The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k
colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each …

We give a new, stronger proof that there are only finitely many k-vertex-critical (P 5, gem)-
free graphs for all k. Our proof further refines the structure of these graphs and allows for the …

A graph is called P t-free if it does not contain the path on t vertices as an induced subgraph.
Let H be a multigraph with the property that any two distinct vertices share at most one …

The Colouring problem is to decide if the vertices of a graph can be coloured with at most k
colours for an integer k, such that no two adjacent vertices are coloured alike. A graph G is H …

A Hamiltonian path (a Hamiltonian cycle) in a graph is a path (a cycle, respectively) that
traverses all of its vertices. The problems of deciding their existence in an input graph are …

We study the Max Partial h-Coloring problem: given a graph G, find the largest induced
subgraph of G that admits a homomorphism into H, where H is a fixed pattern graph without …

A graph G is k-vertex-critical if χ (G)= k but χ (G− v)< k for all v∈ V (G) where χ (G) denotes
the chromatic number of G. We show that there are only finitely many k-vertex-critical (P 3+ ℓ …

Let P k be a path, C ka cycle on k vertices, and K k, ka complete bipartite graph with k
vertices on each side of the bipartition. We prove that (1) for any integers k, t> 0 and a graph …

For graphs $ G $ and $ H $, an $ H $-coloring of $ G $ is an edge-preserving mapping from
$ V (G) $ to $ V (H) $. In the $ H $-Coloring problem the graph $ H $ is fixed and we ask …