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Proportional Choosability of Complete Bipartite Graphs, It Is Worth Briefly Addressing the Choosability of Complete Bipartite Graphs

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Proportional Choosability of Complete Bipartite Graphs, It Is Worth Briefly Addressing the Choosability of Complete Bipartite Graphs Proportional Choosability of Complete Bipartite Graphs Jeffrey A. Mudrock,∗ Jade Hewitt,∗ Paul Shin,∗ and Collin Smith∗ May 28, 2020 Abstract Proportional choosability is a list analogue of equitable coloring that was introduced in 2019. The smallest k for which a graph G is proportionally k-choosable is the proportional choice number of G, and it is denoted χpc(G). In the first ever paper on proportional choosability, it was shown that when 2 ≤ n ≤ m, max{n +1, 1+ ⌈m/2⌉} ≤ χpc(Kn,m) ≤ n + m − 1. In this note we improve on this result by showing that max{n +1, ⌈n/2⌉ + ⌈m/2⌉} ≤ χpc(Kn,m) ≤ n + m − 1 −⌊m/3⌋. In the process, we prove some new lower bounds on the proportional choice number of complete multipartite graphs. We also present several interesting open questions. Keywords. graph coloring, equitable coloring, list coloring, proportional choosability Mathematics Subject Classification. 05C15 1 Introduction In this note all graphs are nonempty, finite, simple graphs unless otherwise noted. Gener- ally speaking we follow West [24] for terminology and notation. The set of natural numbers is N = {1, 2, 3,...}. For m ∈ N, we write [m] for the set {1,...,m}. If G is a graph and S ⊆ V (G), we use G[S] for the subgraph of G induced by S. We also let NG(S) be the subset of V (G) consisting of all vertices adjacent (in G) to at least one vertex in S. We write ∆(G) for the maximum degree of a vertex in G. We write Kn,m for the equivalence class consisting of complete bipartite graphs with partite sets of size n and m, and we write Kn1,...,nt for the equivalence class consisting of complete t-partite graphs with partite sets of size n1,...,nt. arXiv:2005.12915v1 [math.CO] 26 May 2020 In 2019, a new notion combining the notions of equitable coloring and list coloring called proportional choosability was introduced [10]. In this note, we study the proportional choos- ability of complete bipartite graphs. Before introducing proportional choosability, we briefly review equitable coloring and list coloring. 1.1 Equitable Coloring and List Coloring In the classical vertex coloring problem, we seek to color the vertices of a graph with up to k colors so that adjacent vertices receive different colors, a so-called proper k-coloring. The chromatic number of a graph G, denoted χ(G), is the smallest k such that G has a proper 1Department of Mathematics, College of Lake County, Grayslake, IL 60030. E-mail: [email protected] 1 k-coloring. Equitable coloring is a variation on the classical vertex coloring problem that began with a conjecture of Erd˝os [3] in 1964. This conjecture of Erd˝os was proved in 1970 [6] (Theorem 1 below). In 1973, Meyer [15] formally introduced equitable coloring. An equitable k-coloring of a graph G is a proper k-coloring f of G such that the sizes of the color classes associated with f differ by at most one. 1 It is easy to see that for an equitable k-coloring, the color classes associated with the coloring are each of size ⌈|V (G)|/k⌉ or ⌊|V (G)|/k⌋. We say that a graph G is equitably k-colorable if there exists an equitable k-coloring of G. Equitable colorings are useful when it is preferable to form a proper coloring without under-using or over-using any color (see [8, 9, 21, 22] for applications). Unlike ordinary graph coloring, increasing the number of colors can make equitable col- oring more difficult. For example, K3,3 is equitably 2-colorable, but it is not equitably 3-colorable. In 1970, Hajn´al and Szemer´edi proved the following result. Theorem 1 ([6]). Every graph G has an equitable k-coloring when k ≥ ∆(G) + 1. List coloring is another well-known variation on the classical vertex coloring problem, and it was introduced independently by Vizing [23] and Erd˝os, Rubin, and Taylor [4] in the 1970s. For list coloring, we associate a list assignment L with a graph G such that each vertex v ∈ V (G) is assigned a list of colors L(v) (we say L is a list assignment for G). The graph G is L-colorable if there exists a proper coloring f of G such that f(v) ∈ L(v) for each v ∈ V (G) (we refer to f as a proper L-coloring of G). A list assignment L is called a k-assignment for G if |L(v)| = k for each v ∈ V (G). The list chromatic number of a graph G, denoted χℓ(G), is the smallest k such that G is L-colorable whenever L is a k-assignment for G. We say G is k-choosable if k ≥ χℓ(G). Suppose that L is a list assignment for a graph G. The palette of colors associated with L is ∪v∈V (G)L(v). From this point forward, we use L to denote the palette of colors associated with L whenever L is a list assignment. We say that L is a constant k-assignment for G when L is a k-assignment for G and |L| = k (i.e., L assigns the same list of k colors to every vertex in V (G)). Since G must be L-colorable whenever L is a constant χℓ(G)-assignment for G, it is clear that χ(G) ≤ χℓ(G). Since the focus of this note is the proportional choosability of complete bipartite graphs, it is worth briefly addressing the choosability of complete bipartite graphs. First, it is well- known that complete bipartite graphs can be used to demonstrate that the list chromatic number and chromatic number of a graph can be arbitrarily far apart. Indeed, χ(Kn,m) = 2 n and χℓ(Kn,m) = n + 1 whenever m ≥ n . It is also known that (1/2 − o(1)) log2(n) ≤ χℓ(Kn,n) ≤ 2 + log2(n) where the o(1)-term tends to zero as n tends to infinity (see [1] and [2]). On the other hand, finding the exact list chromatic number of complete bipartite graphs is notoriously difficult. In fact, the 3-choosable complete bipartite graphs were not fully characterized until 1995 [20] about 20 years after the introduction of list coloring (see section 8.2 in [25] for some discussion). Finally, there are many examples in the literature that study variants of list coloring and equitable variants of list coloring on complete bipartite graphs (see e.g., [5, 12, 16, 17, 18]). 1We work under the assumption that a proper k-coloring has exactly k, possibly empty, color classes. 2 1.2 Proportional Choosability Kostochka, Pelsmajer, and West [13] introduced a list analogue of equitable coloring called equitable choosability in 2003, and this notion has received attention in the literature. If L is a k-assignment for a graph G, a proper L-coloring of G is an equitable L-coloring of G if each color in L appears on at most ⌈|V (G)|/k⌉ vertices. We say G is equitably k-choosable if an equitable L-coloring of G exists whenever L is a k-assignment for G. Importantly, notice that the definition of equitable L-coloring does not place a lower bound on how many times a color must be used, whereas in an equitable k-coloring there is both an upper and lower bound (i.e., ⌈|V (G)|/k⌉ and ⌊|V (G)|/k⌋) on how many times each color must be used. Kaul, Pelsmajer, Reiniger, and the first author [10] introduced a new list analogue of equitable coloring called proportional choosability which places both an upper and lower bound on how many times a color must be used in a list coloring. The study of proportional choosability is in its infancy, but it has received some attention in the literature (see [10, 11, 19]). Now, following [10] we introduce the specifics. Suppose that L is a k-assignment for a graph G. For each color c ∈ L, the multiplicity of c in L is the number of vertices v whose list L(v) contains c. The multiplicity of c in L is denoted by ηL(c) (or simply η(c) when the list assignment is clear). So, ηL(c)= |{v ∈ V (G) : c ∈ L(v)}|. A proper L-coloring f of G is a proportional L-coloring of G if for each c ∈L, f −1(c), the color class of c, is of size η(c) η(c) or . k k We say G is proportionally L-colorable if a proportional L-coloring of G exists, and we say G is proportionally k-choosable if G is proportionally L-colorable whenever L is a k-assignment for G. Proportional choosability has some notable (perhaps surprising) properties. Proposition 2 ([10]). If G is proportionally k-choosable, then G is both equitably k-choosable and equitably k-colorable. Proposition 3 ([10]). If G is proportionally k-choosable, then G is proportionally (k + 1)- choosable. The proportional choice number of a graph G, denoted χpc(G), is the smallest integer k such that G is proportionally k-choosable. In light of Proposition 3, we know that a graph is proportionally k-choosable if and only if k ≥ χpc(G). Proposition 4 ([10]). Suppose H is a subgraph of G. If G is proportionally k-choosable, then H is proportionally k-choosable. Consequently, χpc(G) ≥ χpc(H). Notice that Propositions 3 and 4 are particularly interesting since they do not hold in the contexts of equitable coloring and equitable choosability.
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