Choice numbers of graphs

A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing that if a graph $G$ is $(a:b)$-choosable, and $c/d > a/b$, then $G$ is not necessarily $(c:d)$-choosable. The simplest case of another problem, stated by the same authors, is settled, proving that every 2-choosable graph is also $(4:2)$-choosable. Applying probabilistic methods, an upper bound for the $k^{th}$ choice number of a graph is given. We also prove that a directed graph with maximum outdegree $d$ and no odd directed cycle is $(k(d+1):k)$-choosable for every $k \geq 1$. Other results presented in this article are related to the strong choice number of graphs (a generalization of the strong chromatic number). We conclude with complexity analysis of some decision problems related to graph choosability.

💡 Research Summary

The paper investigates the generalized list‑coloring concept known as (a : b)‑choosability and establishes several new results that deepen our understanding of choice numbers in graphs. After recalling the definition—each vertex receives a list of a colors and must be assigned b of them so that adjacent vertices receive disjoint sets—the author focuses on three main themes: asymptotic bounds for the k‑th choice number, structural relations between different (a : b) parameters, and algorithmic/complexity aspects.

The first major theorem shows that for any graph G and any ε > 0 there exists a threshold k₀ such that for all k ≥ k₀ the k‑th choice number satisfies chₖ(G) ≤ k·(χ(G)+ε). The proof uses a probabilistic method: each vertex is given ⌊k(χ(G)+ε)⌋ colors, a random function maps each color to one of the χ(G) color classes, and Chernoff‑type bounds guarantee that with positive probability every vertex retains at least k colors in its own class. Consequently, for large k the choice number grows essentially linearly with the chromatic number, improving earlier logarithmic‑type bounds.

Using this result the author answers negatively a question of Erdős, Rubin and Taylor: does (a : b)‑choosability imply (c : d)‑choosability whenever c/d > a/b? By constructing graphs with prescribed choice number l+1 and chromatic number m−1 (for any l > m ≥ 3) and applying the previous theorem, it is shown that such a graph can be (k·m : k)‑choosable but not (l : 1)‑choosable, disproving the conjectured monotonicity.

The second part derives explicit upper bounds for the k‑th choice number of complete multipartite graphs. Lemma 3.2 handles the case r ≤ t (where t is the sum of part sizes) and yields chₖ ≤ 4r(k+log t). For r > t a recursive partitioning argument together with refined constants leads to chₖ ≤ 244 r(k+log t). Combining both cases and optimizing constants gives the clean bound

  chₖ(K_{m₁,…,mᵣ}) ≤ 948 r (k + log m₁ + … + log mᵣ).

From this, Corollary 1.4 follows: for any graph G, chₖ(G) ≤ 948 χ(G)·(k + log(|V|·χ(G)+1)). The author also applies the result to random graphs G_{n,p}, showing that with high probability chₖ(G_{n,p}) ≤ 475 log(1/(1−p))·n·log log n·log n, extending known results on ordinary choice numbers.

The third theme connects orientations to choosability. Theorem 1.6 proves that if a digraph D has maximum out‑degree d and contains no odd directed cycle, then assigning each vertex a list of size k(d+outdeg(v)+1) guarantees a selection of k colors per vertex with pairwise disjoint choices on each edge. Moreover, a polynomial‑time algorithm constructs such a selection. Consequently, any undirected graph admitting an orientation with out‑degree at most d and no odd directed cycles is (k(d+1) : k)‑choosable (Corollary 1.7). This yields several corollaries: even cycles are (2k : k)‑choosable, Brooks‑type bounds chₖ(G) ≤ k·Δ(G) for connected non‑complete, non‑odd‑cycle graphs, and specific bounds for bipartite and planar bipartite graphs.

The fourth section returns to the classic problem of 2‑choosability. Using the structural characterization of 2‑choosable graphs (cores are K₁, even cycles, or Θ‑graphs), Theorem 1.16 shows that every 2‑choosable graph is also (4 : 2)‑choosable, providing a partial positive answer to the question “does (a : b)‑choosable imply (am : bm)‑choosable?”. Moreover, Theorem 1.17 establishes that for odd k, (2mk : mk)‑choosability implies 2m‑choosability.

The fifth part studies computational complexity. It is known that Bipartite (2,3)‑choosability is Π₂^p‑complete. Theorem 1.18 extends this hardness: for any fixed integer k ≥ 3, deciding whether a bipartite graph is k‑choosable is Π₂^p‑complete, while the case k = 2 is polynomial‑time solvable (by the earlier structural theorem).

Finally, the paper introduces the notion of strong choosability (analogous to strong chromatic number). A graph is strongly k‑choosable if after adding any collection of vertex‑disjoint cliques of size at most k the resulting graph remains k‑choosable. Theorem 1.19 proves that strong k‑choosability implies strong (k+1)‑choosability, and Theorem 1.20 shows strong k‑choosability ⇒ strong km‑choosability. Theorem 1.23 improves the lower bound on the strong choice number of maximum degree d graphs, proving sχ(d) ≥ 2d, which doubles the previously known bound.

In summary, the article delivers a comprehensive treatment of (a : b)‑choosability: it provides asymptotically tight linear bounds for large k, disproves a natural monotonicity conjecture, links orientations without odd cycles to (k(d+1) : k)‑choosability, settles the (4 : 2) case for 2‑choosable graphs, establishes Π₂^p‑completeness for bipartite k‑choosability (k ≥ 3), and advances the theory of strong choosability. These contributions significantly extend the toolbox for researchers studying list coloring, choice numbers, and related combinatorial optimization problems.