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.
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