An independent transversal of a graph G with a vertex partition  is an independent set of G intersecting each block of  in a single vertex. Wanless and Wood proved that if each block of  has size at least t and the average degree of vertices in each block is at most t 4 ∕ , then an independent transversal of  exists. We present a construction showing that this result is optimal: for any ε > 0 and sufficiently large t, there is a forest with a vertex partition into parts of size at least t such that the average degree of vertices in each block is, and there is no independent transversal. This unexpectedly shows that methods related to entropy compression such as the Rosenfeld-Wanless-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.

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A Gallai coloring is an edge coloring that avoids triangles colored with three different colors. Given integers e 1 ≥ e 2 ≥ . . . ≥ e k with k i=1 e i = n 2 for some n, does there exist a Gallai k-coloring of K n with e i edges in color i? In this paper, we give several sufficient conditions and one necessary condition to guarantee a positive answer to the above question. In particular, we prove the existence of a Gallai-coloring if e 1 − e k ≤ 1 and k ≤ ⌊n/2⌋. We prove that for any integer k ≥ 3 there is a (unique) integer g(k) with the following property: there exists a Gallai k-coloring of K n with e i edges in color i for every e 1 ≤ . . . ≤ e k satisfying k i=1 e i = n 2 , if and only if n ≥ g(k). We show that g(3) = 5, g(4) = 8, and 2k − 2 ≤ g(k) ≤ 8k 2 + 1 for every k ≥ 3.

An independent transversal of a graph G with a vertex partition P is an independent set of G intersecting each block of P in a single vertex. Wanless and Wood proved that if each block of 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 P exists. We present a construction showing that this result is optimal: for any ε > 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 ( 1 4 + ε)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.