3 research outputs found
A Simple Characterization of Proportionally 2-choosable Graphs
We recently introduced proportional choosability, a new list analogue of
equitable coloring. Like equitable coloring, and unlike list equitable coloring
(a.k.a. equitable choosability), proportional choosability bounds sizes of
color classes both from above and from below. In this note, we show that a
graph is proportionally 2-choosable if and only if it is a linear forest such
that its largest component has at most 5 vertices and all of its other
components have two or fewer vertices. We also construct examples that show
that characterizing equitably 2-choosable graphs is still open.Comment: 9 page
Proportional Choosability of Complete Bipartite Graphs
Proportional choosability is a list analogue of equitable coloring that was
introduced in 2019. The smallest for which a graph is proportionally
-choosable is the proportional choice number of , and it is denoted
. In the first ever paper on proportional choosability, it was
shown that when , . In this note we improve on this result
by showing that . 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.Comment: 11 page
Proportional 2-Choosability with a Bounded Palette
Proportional choosability is a list coloring analogue of equitable coloring.
Specifically, a -assignment for a graph specifies a list of
available colors to each . An -coloring assigns a color to
each vertex from its list . A proportional -coloring of is a
proper -coloring in which each color is
used or times where
. A graph is
proportionally -choosable if a proportional -coloring of exists
whenever is a -assignment for . Motivated by earlier work, we
initiate the study of proportional choosability with a bounded palette by
studying proportional 2-choosability with a bounded palette. In particular,
when , a graph is said to be proportionally -choosable if a proportional -coloring of exists whenever is a
-assignment for satisfying . We
observe that a graph is proportionally -choosable if and only if it is
equitably 2-colorable. As gets larger, the set of proportionally -choosable graphs gets smaller. We show that whenever a
graph is proportionally -choosable if and only if it is
proportionally 2-choosable. We also completely characterize the connected
proportionally -choosable graphs when .Comment: 20 page