Andrzej Grzesik

Indicated coloring of graphs

Abstract We study a graph coloring game in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent realization of this project. The smallest number of colors necessary for Ann to win the game on a graph G (regardless of Ben’s strategy) is called the indicated chromatic number of G , and denoted by χ i ( G ) . We approach the question how much χ i ( G ) differs from the usual chromatic number χ ( G ) . In particular, whether there is a function f such that χ i ( G ) ⩽ f ( χ ( G ) ) for every graph G . We prove that f cannot be linear with leading coefficient less than 4 / 3 . On the other hand, we show that the indicated chromatic number of random graphs is bounded roughly by 4 χ ( G ) . We also exhibit several classes of graphs for which χ i ( G ) = χ ( G ) and show that this equality for any class of perfect graphs implies Clique-Pair Conjecture for this class of graphs.

Interval edge-colorings of K1,m,n

In this note we prove that K_{1,m,n} is interval edge-colorable if and only if gcd(m+1,n+1)=1. It settles in the affirmative a conjecture of Petrosyan.

From Directed Path to Linear Order---The Best Choice Problem for Powers of Directed Path

We examine the evolution of the best choice algorithm and the probability of its success from a directed path to the linear order of the same cardinality through

Avoider-enforcer star games

k

Extremal graph theory and finite forcibility

th powers of a directed path,

Interval edge-colorings of K1,m,nK1,m,n

1...

On the maximum number of five-cycles in a triangle-free graph

In this paper, we study (1 : b) Avoider-Enforcer games played on the edge set of the complete graph on n vertices. For every constant k≥3 we analyse the k-star game, where Avoider tries to avoid claiming k edges incident to the same vertex. We consider both versions of Avoider-Enforcer games — the strict and the monotone — and for each provide explicit winning strategies for both players. We determine the order of magnitude of the threshold biases fmonF, f-F and f+F, where F is the hypergraph of the game.

Finitely forcible graphons and permutons

Abstract We study the uniqueness of optimal solutions to extremal graph theory problems. Our main result is a counterexample to the following conjecture of Lovasz, which is often referred to as saying that “every extremal graph theory problem has a finitely forcible optimum”: every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints such that the resulting set is satisfied by an asymptotically unique graph.

Minimum number of edges that occur in odd cycles.

In this note we prove that K_{1,m,n} is interval edge-colorable if and only if gcd(m+1,n+1)=1. It settles in the affirmative a conjecture of Petrosyan.