Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Note on the Equitable Choosability of Complete Bipartite Graphs   In 2003 Kostochka, Pelsmajer, and West introduced a list analogue of
equitable coloring called equitable choosability. A $k$-assignment, $L$, for a
graph $G$ assigns a list, $L(v)$, of $k$ available colors to each $v \in V(G)$,
and an equitable $L$-coloring of $G$ is a proper coloring, $f$, of $G$ such
that $f(v) \in L(v)$ for each $v \in V(G)$ and each color class of $f$ has size
at most $\lceil |V(G)|/k \rceil$. Graph $G$ is said to be equitably
$k$-choosable if an equitable $L$-coloring of $G$ exists whenever $L$ is a
$k$-assignment for $G$. In this note we study the equitable choosability of
complete bipartite graphs. A result of Kostochka, Pelsmajer, and West implies
$K_{n,m}$ is equitably $k$-choosable if $k \geq \max \{n,m\}$ provided $K_{n,m}
\neq K_{2l+1, 2l+1}$. We prove $K_{n,m}$ is equitably $k$-choosable if $m \leq
\left\lceil (m+n)/k \right \rceil(k-n)$ which gives $K_{n,m}$ is equitably
$k$-choosable for certain $k$ satisfying $k < \max \{n,m\}$. We also give a
complete characterization of the equitable choosability of complete bipartite
graphs that have a partite set of size at most 2.

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The categories of this paper are: q-bio
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The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: stat
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The categories of this paper are: plasm-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int
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The categories of this paper are: quant-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: cs