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.
Deep Dive into 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.
A graph G = (V, E) is (a : b)-choosable if for every family of sets {S(v) : v ∈ V }, where |S(v)| = a for all v ∈ V , there are subsets C(v) ⊆ S(v), where |C(v)| = b for all v ∈ V , and C(u) ∩ C(v) = ∅ for every two adjacent vertices u, v ∈ V . The kth choice number of G, denoted by ch k (G), is the minimum integer n so that G is (n : k)-choosable. A graph G = (V, E) is k-choosable if it is (k : 1)-choosable. The choice number of G, denoted by ch(G), is equal to ch 1 (G).
The concept of (a : b)-choosability was defined and studied by Erdős, Rubin and Taylor in [8].
In the present paper we prove several results concerning (a : b)-choosability, a number of which generalize known results regarding choice numbers of graphs that appear in [4] and [2]. The following theorem examines the behavior of ch k (G) when k is large.
Theorem 1.1 Let G be a graph. For every ǫ > 0 there exists an integer k 0 such that ch k (G) ≤ k(χ(G) + ǫ) for every k ≥ k 0 .
In [8] the authors ask the following question:
If G is (a : b)-choosable, and c d > a b , does it follow that G is (c : d)-choosable? The following corollary gives a negative answer to this question. Let K m * r denote the complete r-partite graph with m vertices in each vertex class, and let K m 1 ,…,mr denote the complete r-partite graph with m i vertices in the ith vertex class. It is shown in [2] that there exist two positive constants c 1 and c 2 such that for every m ≥ 2 and for every r ≥ 2, c 1 r log m ≤ ch(K m * r ) ≤ c 2 r log m. The following theorem generalizes the upper bound.
Theorem 1.3 If r ≥ 1 and m i ≥ 2 for every i, 1 ≤ i ≤ r, then ch k (K m 1 ,…,mr ) ≤ 948r(k + log
The following are two applications of this theorem.
Corollary 1.4 For every graph G and k ≥ 1
The second corollary generalizes a result from [2] concerning the choice numbers of random graphs for the common model G n,p (see, e.g., [7]), in which the graph is obtained by taking each pair of the n labeled vertices 1, 2, . . . , n to be an edge, randomly and independently, with probability p.
Corollary 1.5 For every two constants k ≥ 1 and 0 < p < 1, the probability that ch k (G n,p ) ≤ 475 log (1/(1p))n log log n log n tends to 1 as n tends to infinity.
A theorem which appears in [4] reveals the connection between the choice number of a graph G and its orientations. We present here a generalization of this theorem for a special case. Corollary 1.8 An even cycle is (2k : k)-choosable for every k ≥ 1.
The last corollary enables us to prove a generalization of a variant of Brooks Theorem which appears in [8].
Corollary 1.9 If a connected graph G is not K n , and not an odd cycle, then ch k (G) ≤ k∆(G) for every k ≥ 1, where ∆(G) is the maximum degree of G.
For a graph G = (V, E), define M (G) = max(|E(H)|/|V (H)|), where H = (V (H), E(H)) ranges over all subgraphs of G. The following two corollaries are generalizations of results which appear in [4]. The following are additional applications.
Corollary 1.12 If every induced subgraph of a graph G has a vertex of degree at most d, then G
where ω(G) is the clique number of G.
The list-chromatic conjecture asserts that for every graph
is the line graph of G. The list-chromatic conjecture is easy to establish for trees, graphs of degree at most 2, and K 2,m . It has also been verified for snarks [11], K 3,3 , K 4,4 , K 6,6 [4], and 2-connected cubic planar graphs. The following corollary shows that the list-chromatic conjecture is true for graphs which contain no C n for every n ≥ 4.
The core of a graph G is the graph obtained from G by deleting nodes of degree 1 successively until there are no nodes of degree 1. The graph Θ a,b,c consists of two distinguished nodes u and v together with three paths of lengths a,b, and c, which are node disjoint except that each path has u at one end, and v at the other end. The following theorem from [8] gives a characterization of the 2-choosable graph:
In [8] the authors ask the following question:
The following theorem gives a partial solution to this question by using theorem 1.15.
Theorem 1.17 Suppose that k and m are positive integers and that k is odd. If a graph G is
there is a proper vertex-coloring of G assigning to each vertex v ∈ V a color from S(v). It is shown in [8] that the following problem is Π p 2 -complete: ( for terminology see [10] )
We consider the following decision problem:
If follows from theorem 1.15 that this problem is solvable in polynomial time for k = 2.
A graph G = (V, E) is strongly k-colorable if every graph obtained from G by adding to it a union of vertex disjoint cliques of size at most k ( on the set V ) is k-colorable. An analogous definition of strongly k-choosable is made by replacing colorability with choosability. The strong chromatic number of a graph G, denoted by sχ(G), is the minimum k such that G is strongly k-colorable. Define sχ(d) = max(sχ(G)), where G ranges over all graphs with maximum degree at most d. The definition of strongly k-
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
