Ágnes Tóth

Permutation Capacities of Families of Oriented Infinite Paths

Korner and Malvenuto asked whether one can find

Gallai colorings and domination in multipartite digraphs

\binom{n}{\lfloor n/2\rfloor}

The Ultimate Categorical Independence Ratio of Complete Multipartite Graphs

linear orderings (i.e., permutations) of the first

Permutation Capacities and Oriented Infinite Paths

n

Around a biclique cover conjecture

natural numbers such that any pair of them places two consecutive integers somewhere in the same position. This led to the notion of graph-different permutations. We extend this concept to directed graphs, focusing on orientations of the semi-infinite path whose edges connect consecutive natural numbers. Our main result shows that the maximum number of permutations satisfying all the pairwise conditions associated with all of the various orientations of this path is exponentially smaller, for any single orientation, than the maximum number of those permutations which satisfy the corresponding pairwise relationship. This is in sharp contrast to a result of Gargano, Korner, and Vaccaro concerning the analogous notion of Sperner capacity of families of finite graphs. We improve the exponential lower bound for the original problem and list a number of open questions.

Partition of Graphs and Hypergraphs into Monochromatic Connected Parts

Assume that D is a digraph without cyclic triangles and its vertices are partitioned into classes A1, …, At of independent vertices. A set is called a dominating set of size |S| if for any vertex there is a w∈U such that (w, v)∈E(D). Let β(D) be the cardinality of the largest independent set of D whose vertices are from different partite classes of D. Our main result says that there exists a h = h(β(D)) such that D has a dominating set of size at most h. This result is applied to settle a problem related to generalized Gallai colorings, edge colorings of graphs without 3-colored triangles.

Answer to a question of Alon and Lubetzky about the ultimate categorical independence ratio

The independence ratio

On the ultimate lexicographic Hall-ratio

i(G)

On the ultimate categorical independence ratio

of a graph

Asymptotic values of graph parameters

G