Algorithms for Weighted Independent Transversals and Strong Colouring

An independent transversal (IT) in a graph with a given vertex partition is an independent set consisting of one vertex in each partition class. Several sufficient conditions are known for the existence of an IT in a given graph and vertex partition, which have been used over the years to solve many combinatorial problems. Some of these IT existence theorems have algorithmic proofs, but there remains a gap between the best existential bounds and the bounds obtainable by efficient algorithms. Recently, Graf and Haxell (2018) described a new (deterministic) algorithm that asymptotically closes this gap, but there are limitations on its applicability. In this article, we develop a randomized algorithm that is much more widely applicable, and demonstrate its use by giving efficient algorithms for two problems concerning the strong chromatic number of graphs.

Partitioning the Power Set of $[n]$ into $C_k$-free parts

We show that for $n \geq 3, n\ne 5$, in any partition of $\mathcal{P}(n)$, the set of all subsets of $[n]=\{1,2,\dots,n\}$, into $2^{n-2}-1$ parts, some part must contain a triangle — three different subsets $A,B,C\subseteq [n]$ such that $A\cap B,A\cap C,B\cap C$ have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into $2^{n-2}$ triangle-free parts. We also address a more general Ramsey-type problem: for a given graph $G$, find (estimate) $f(n,G)$, the smallest number of colors needed for a coloring of $\mathcal{P}(n)$, such that no color class contains a Berge-$G$ subhypergraph. We give an upper bound for $f(n,G)$ for any connected graph $G$ which is asymptotically sharp when $G$ is a cycle, path, or star. Additional bounds are given when $G$ is a $4$-cycle and when $G$ is a claw.

Locating Chromatic Number of Powers of Paths and Cycles

Let c be a proper k-coloring of a graph G. Let π = { R 1 , R 2 , … , R k } be the partition of V ( G ) induced by c, where R i is the partition class receiving color i. The color code c π ( v ) of a vertex v of G is the ordered k-tuple ( d ( v , R 1 ) , d ( v , R 2 ) , … , d ( v , R k ) ) , where d ( v , R i ) is the minimum distance from v to each other vertex u ∈ R i for 1 ≤ i ≤ k . If all vertices of G have distinct color codes, then c is called a locating k-coloring of G. The locating-chromatic number of G, denoted by χ L ( G ) , is the smallest k such that G admits a locating coloring with k colors. In this paper, we give a characterization of the locating chromatic number of powers of paths. In addition, we find sharp upper and lower bounds for the locating chromatic number of powers of cycles.

On the Locating Chromatic Number of Certain Barbell Graphs

The locating chromatic number of a graph G is defined as the cardinality of a minimum resolving partition of the vertex set V(G) such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices in G are not contained in the same partition class. In this case, the coordinate of a vertex v in G is expressed in terms of the distances of v to all partition classes. This concept is a special case of the graph partition dimension notion. In this paper we investigate the locating chromatic number for two families of barbell graphs.

A Probabilistic Counting Lemma for Complete Graphs

International audience We prove the existence of many complete graphs in almost all sufficiently dense partitions obtained by an application of Szemerédi's Regularity Lemma. More precisely, we consider the number of complete graphs $K_{\ell}$ on $\ell$ vertices in $\ell$-partite graphs where each partition class consists of $n$ vertices and there is an $\varepsilon$-regular graph on $m$ edges between any two partition classes. We show that for all $\beta > $0, at most a $\beta^m$-fraction of graphs in this family contain less than the expected number of copies of $K_{\ell}$ provided $\varepsilon$ is sufficiently small and $m \geq Cn^{2-1/(\ell-1)}$ for a constant $C > 0$ and $n$ sufficiently large. This result is a counting version of a restricted version of a conjecture by Kohayakawa, Łuczak and Rödl and has several implications for random graphs.

A Baire-type theorem for cardinals

We shall prove the following theorem: If κ is an infinite cardinal and Γ: P(κ) → cf κ a partition of the power-set of κ then we can find a homogeneous ⊿-system of size κ such that the kernel of the ⊿-system is in the same partition class as all the members of the ⊿-system.