An independent transversal of a graph $G$ with a vertex partition $\mathcal P$ is an independent set of $G$ intersecting each block of $\mathcal P$ in a single vertex. Wanless and Wood proved that if each block of $\mathcal P$ has size at least $t$ and the average degree of vertices in each block is at most $t/4$, then an independent transversal of $\mathcal P$ exists. We present a construction showing that this result is optimal: for any $\varepsilon > 0$ and sufficiently large $t$, there is a family of forests with vertex partitions whose block size is at least $t$, average degree of vertices in each block is at most $(\frac14+\varepsilon)t$, and there is no independent transversal. This unexpectedly shows that methods related to entropy compression such as the Rosenfeld-Wanles-Wood scheme or the Local Cut Lemma are tight for this problem. Further constructions are given for variants of the problem, including the hypergraph version.

Graphs of low average degree without independent transversals

Threshold Ramsey multiplicity for odd cycles

Abstract List coloring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list‐coloring, we seek many in parallel. Our explorations have uncovered a potentially rich seam of interesting problems spanning chromatic graph theory. Given a ‐list‐assignment of a graph , which is the assignment of a list of colors to each vertex , we study the existence of pairwise‐disjoint proper colorings of using colors from these lists. We may refer to this as a list‐packing . Using a mix of combinatorial and probabilistic methods, we set out some basic upper bounds on the smallest for which such a list‐packing is always guaranteed, in terms of the number of vertices, the degeneracy, the maximum degree, or the (list) chromatic number of . (The reader might already find it interesting that such a minimal is well defined.) We also pursue a more focused study of the case when is a bipartite graph. Our results do not yet rule out the tantalising prospect that the minimal above is not too much larger than the list chromatic number. Our study has taken inspiration from study of the strong chromatic number, and we also explore generalizations of the problem above in the same spirit. 

Packing list‐colorings

Given a graph $G$ and a partition $P$ of its vertex set, an independent transversal (IT) is an independent set of $G$ that contains one vertex from each block in $P$. Various sufficient conditions for the existence of an IT have been established, and a common theme for many of them is that the block sizes are sufficiently large compared to the maximum degree of $G$. Consequently, there has been interest in constructing graphs with no IT which demonstrate that these bounds on the block sizes are best possible. We describe a simple systematic method for constructing vertex-partitioned graphs with large block sizes and no IT. Unifying previous constructions, we use our method to derive classical extremal constructions due to Jin (1992), Yuster (1997), and Szabó and Tardos (2006) in streamlined fashion. For our new results, we describe extremal constructions of minimal graphs with maximum degree two and no IT, generalizing a result of Aharoni, Holzman, Howard, and Sprüssel (2015). We construct a family of locally sparse graphs with no IT, complementing an asymptotic result of Loh and Sudakov (2007). We describe new and smaller counterexamples to a list coloring conjecture of Reed (1999), which was originally disproved by Bohman and Holzman (2002). We disprove a conjecture of Aharoni, Alon, and Berger (2016) about IT's in graphs without large induced stars. We answer negatively a question of Aharoni, Holzman, Howard, and Sprüssel (2015) about extremal graphs with no IT, but we also prove that a useful variation of their question does hold. 

pdf/constructing-graphs-with-no-independent-transversals-19tadficuy.pdf

Constructing Graphs with No Independent Transversals

The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum number of monochromatic copies of $H$ taken over all two-edge-colorings of $K_{r(H)}$. The study of this concept was first proposed by Harary and Prins almost fifty years ago. In a companion paper, the authors have shown that there is a positive constant $c$ such that the threshold Ramsey multiplicity for a path or even cycle with $k$ vertices is at least $(ck)^k$, which is tight up to the value of $c$. Here, using different methods, we show that the same result also holds for odd cycles with $k$ vertices.

The Ramsey number $r(H)$ of a graph $H$ is the minimum integer $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. While this definition only asks for a single monochromatic copy of $H$, it is often the case that every two-edge-coloring of the complete graph on $r(H)$ vertices contains many monochromatic copies of $H$. The minimum number of such copies over all two-colorings of $K_{r(H)}$ will be referred to as the threshold Ramsey multiplicity of $H$. Addressing a problem of Harary and Prins, who were the first to systematically study this quantity, we show that there is a positive constant $c$ such that the threshold Ramsey multiplicity of a path or an even cycle on $k$ vertices is at least $(ck)^k$. This bound is tight up to the constant $c$. We prove a similar result for odd cycles in a companion paper.

Threshold Ramsey multiplicity for paths and even cycles

/pdf/threshold-ramsey-multiplicity-for-paths-and-even-cycles-18epj6sq.pdf

An average degree condition for independent transversals

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Threshold Ramsey multiplicity for paths and even cycles

Threshold Ramsey multiplicity for paths and even cycles

An average degree condition for independent transversals