Multiple list coloring of 3‐choice critical graphs

Abstract

A graph G is called 3-choice critical if G is not 2-choosable but any proper subgraph is 2-choosable. A characterization of 3-choice critical graphs was given by Voigt in 1998. Voigt conjectured that if G is a bipartite 3-choice critical graph, then G is m m (4 , 2)choosable for every integer m. This conjecture was disproved by Meng et al. in 2017. They showed that if G = Θ r s t , , where r s t , , have the same parity and r s t min{ , , } 3 ≥ , or G = Θ p 2,2,2,2 with p 2 ≥ , then G is How to cite this article: Xu R, Zhu X. Multiple list coloring of 3-choice critical graphs.

The strong fractional choice number of $3$-choice critical graphs

“…The following is a key lemma for the proof in this paper. It generalizes Lemma 9 in [10], which is a special case where and k = 2m − τ are even. Lemma 2.9 Let and k be fixed integers, where k ≥ 1, > k, 0 ≤ τ ≤ m. Assume x, y are non-negative integers with x + y ≤ .…”

“…The proof of Theorem 1.3 uses the idea in [5,10]: Assume G is a graph as in Theorem 1.3 and L is a (2m + 1)-list assignment of G. Let u, v be the two vertices of G of degree at least 3. Then G − {u, v} is the disjoint union of a family of three or four paths, where each end vertex of these paths has exactly one neighbour in {u, v} unless the path consists of a single vertex w, in which case w is adjacent to both u and v. Other vertices of the paths are not adjacent to u or v.…”

“…For the readability, we suppress the index '+'. The following lemma is proved in [10] (Lemma 18 in [10], where the parameter 2k is changed to k, but the proof still works).…”

“…Nevertheless, the other bipartite 3choice critical graphs, i.e., two vertex-disjoint even cycles joined by a path, two even cycles with one vertex in common, Θ 2,2s,2t with s, t > 1, and Θ 1,2s+1,2t+1 with s, t > 0, are (4, 2)-choosable [5]. Xu and Zhu [10] strengthened these results and proved that these graphs are also (4m, 2m)-choosable for all integer m. Note that if a graph G is (4m, 2m)choosable, then it is (4m − 1, 2m − 1)-choosable: if L is a (4m − 1)-list assignment, then let c be a new colour, and let L ′ (v) = L(v) ∪ {c}, we obtain a 4m-list assignment. Let f be a 2m-fold L ′ -colouring of G, and let…”

The strong fractional choice number of 3‐choice‐critical graphs

“…Nevertheless, the other bipartite 3‐choice‐critical graphs, that is, two vertex‐disjoint even cycles joined by a path, two even cycles with one vertex in common, $${{\rm{\Theta }}}_{2,2s,2t}$$ with $$s,t\gt 1$$, and $${{\rm{\Theta }}}_{1,2s+1,2t+1}$$ with $$s,t\gt 0$$, are $$(4,2)$$‐choosable [5]. Xu and Zhu [10] strengthened these results and proved that these graphs are also $$(4m,2m)$$‐choosable for all integer $$m$$. So $$c{h}_{f}^{s}(G)=2$$ if one of the following holds: …”