Even if there is nothing more to say about the 4-colour-problem, there are very many easily formulated unsolved graph colouring problems left. We have selected a list of 25 pretty problems. If a class of graphs is closed under minors (deletions and contractions), is the maximum chromatic number of graphs in the class equal to the largest order of a complete graph in the class? Assume that all v...

In this paper, we study 3-colorable graphs having no induced 8-vertex path and no induced cycles of specific lengths. We prove a characterization by critical graphs in three particular cases.

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G− v is less than the total domination number of G. These graphs we call γt-critical. If such a graph G has total domination number k, we call it k-γt-critical. We characterize the connected graphs with minimum degree one t...

Cop Robber game is a two player game played on an undirected graph. In this game cops try to capture a robber moving on the vertices of the graph. The cop number of a graph is the least number of cops needed to guarantee that the robber will be caught. In this paper we present results concerning games on G, that is the graph obtained by connecting the corresponding vertices in G and its complem...

Borodin and Kostochka conjectured that every graph G with maximum degree ∆ ≥ 9 satisfies χ ≤ max {ω,∆− 1}. We carry out an in-depth study of minimum counterexamples to the Borodin-Kostochka conjecture. Our main tool is the classification of graph joins A ∗B with |A| ≥ 2, |B| ≥ 2 which are f -choosable, where f(v) := d(v)− 1 for each vertex v. Since such a join cannot be an induced subgraph of a...

A set S of vertices in a graph G = (V,E) is called a total k-distance dominating set if every vertex in V is within distance k of a vertex in S. A graph G is total k-distance domination-critical if γ t (G − x) < γ t (G) for any vertex x ∈ V (G). In this paper, we investigate some results on total k-distance domination-critical of graphs.

A graph is called diameter-k-critical if its diameter is k, and the removal of any edge strictly increases the diameter. In this paper, we prove several results related to a conjecture often attributed to Murty and Simon, regarding the maximum number of edges that any diameter-k-critical graph can have. In particular, we disprove a longstanding conjecture of Caccetta and Häggkvist (that in ever...

A locating-total dominating set of a graph G = (V (G), E(G)) with no isolated vertex is a set S ⊆ V (G) such that every vertex of V (G) is adjacent to a vertex of S and for every pair of distinct vertices u and v in V (G) − S, N(u) ∩ S = N(v) ∩ S. Let γ t (G) be the minimum cardinality of a locating-total dominating set of G. A graph G is said to be locating-total domination vertex critical if ...

A set D of vertices in a connected graph G is called a k-dominating set if every vertex in G − D is within distance k from some vertex of D. The k-domination number of G, γk(G), is the minimum cardinality over all k-dominating sets of G. A graph G is k-distance domination-critical if γk(G − x) < γk(G) for any vertex x in G. This work considers properties of k-distance domination-critical graphs...

We find a lower bound for the proportion of face boundaries of an embedded graph that are nearly–light (that is, they have bounded length and at most one vertex of large degree). As an application, we show that every sufficiently large k–crossing–critical graph has crossing number at most 2k + 23.