A better lower bound on average degree of 4-list-critical graphs

This short note proves that every incomplete $k$-list-critical graph has average degree at least $k-1 + \frac{k-3}{k^2-2k+2}$. This improves the best known bound for $k = 4,5,6$. The same bound holds for online $k$-list-critical graphs.


Introduction
A graph G is k-list-critical if G is not (k − 1)-choosable, but every proper subgraph of G is (k − 1)-choosable. For further definitions and notation, see [5,2]. Table 1 shows some history of lower bounds on the average degree of k-list-critical graphs.
Main Theorem. Every incomplete k-list-critical graph has average degree at least Main Theorem gives a lower bound of 3 + 1 10 for 4-list-critical graphs. This is the first improvement over Gallai's bound of 3 + 1 13 . The same proof shows that Main Theorem holds for online k-list-critical graphs as well. Our primary tool is a lemma proved with Kierstead [6] that generalizes a kernel technique of Kostochka and Yancey [8].
Definition. The maximum independent cover number of a graph G is the maximum mic(G) of I, V (G) \ I over all independent sets I of G.
Kernel Magic (Kierstead and R. [6]). Every k-list-critical graph G satisfies The previous best bounds in Table 1 for k-list-critical graphs hold for k-Alon-Tarsicritical graphs as well. Since Kernel Magic relies on the Kernel Lemma, our proof does not work for k-Alon-Tarsi-critical graphs. Any improvement over Gallai's bound of 3 + 1 13 for 4-Alon-Tarsi-critical graphs would be interesting.  [5] CR [2] Here  Table 1: History of lower bounds on the average degree d(G) of k-critical and k-list-critical graphs G.

The Proof
The connected graphs in which each block is a complete graph or an odd cycle are called Gallai trees. Gallai [4] proved that in a k-critical graph, the vertices of degree k − 1 induce a disjoint union of Gallai trees. The same is true for k-list-critical graphs ( [1,3]). For a graph T and k ∈ N, let β k (T ) be the independence number of the subgraph of T induced on the vertices of degree k − 1. When k is defined in the context, put β(T ) := β k (T ). Proof. Suppose the lemma is false and choose a counterexample T minimizing |T |. Plainly, T has more than one block. Let A be an endblock of T and let x be the unique cutvertex of T with x ∈ V (A). Consider T ′ := T − (V (A) \ {x}). By minimality of |T |, Since T is a counterexample, 2 A > (k − 2)(|A| − 1). So, if k > 4, then A = K k−1 and if k = 4, then A is an odd cycle. In both cases, d T (x) = k − 1. Consider T * := T − V (A). By minimality of |T |, Since T is a counterexample, 2 A + 2 > (k − 2) |A| + 2(β(T ) − β(T * )). In Hence, Let M be the maximum of I, V (G) \ I over all independent sets I of G with I ⊆ H. Then Applying Kernel Magic and using (1) gives Let C be the components of G[H]. Then α(C) ≥ |C| χ(C) for all C ∈ C. Whence If L = ∅, then G has average degree at least k ≥ k − 1 + k−3 k 2 −2k+2 . So, assume L = ∅. Then G[H] is (k − 1)-colorable by k-list-criticality of G. In particular, χ(C) ≤ k − 1 for every C ∈ C. For every C ∈ C, To see this, first suppose C ∈ C is not a tree. Then C ≥ |C| and hence k |C| If C is a tree, then χ(C) ≤ 2 and hence This proves (4) since the bound is trivially satisfied when |C| = 1. Now combining (2), (3) and (4) with the basic bound After some algebra, this becomes That proves the theorem.
The right side of equation (4) in the above proof can be improved to k |C| unless C is a K 2 where both vertices have degree k in G. If these K 2 's could be handled, the average degree bound would improve to k − 1 + k−3 (k−1) 2 .
Conjecture. Every incomplete (online) k-list-critical graph has average degree at least