TY - JOUR
AU - Wagstrom, Tim
AB - Proportional choosability is a list coloring analogue of equitable coloring. Specifically, a k-assignment L for a graph G associates a list L(v) of k available colors to each v∈V(G)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$v \in V(G)$$\end{document}. An L-coloring assigns a color to each vertex v from its list L(v). A proportionalL-coloring of G is a proper L-coloring in which each color c∈⋃v∈V(G)L(v)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$c \in \bigcup _{v \in V(G)} L(v)$$\end{document} is used ⌊η(c)/k⌋\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lfloor \eta (c)/k \rfloor $$\end{document} or ⌈η(c)/k⌉\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lceil \eta (c)/k \rceil $$\end{document} times where η(c)={v∈V(G):c∈L(v)}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\eta (c)=\left|{\{v \in V(G) : c \in L(v) \}}\right|$$\end{document}. A graph G is proportionallyk-choosable if a proportional L-coloring of G exists whenever L is a k-assignment for G. 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 ℓ≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\ell \ge 2$$\end{document}, a graph G is said to be proportionally(2,ℓ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(2, \ell )$$\end{document}-choosable if a proportional L-coloring of G exists whenever L is a 2-assignment for G satisfying |⋃v∈V(G)L(v)|≤ℓ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|\bigcup _{v \in V(G)} L(v)| \le \ell $$\end{document}. We observe that a graph is proportionally (2, 2)-choosable if and only if it is equitably 2-colorable. As ℓ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\ell $$\end{document} gets larger, the set of proportionally (2,ℓ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(2, \ell )$$\end{document}-choosable graphs gets smaller. We show that whenever ℓ≥5\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\ell \ge 5$$\end{document} a graph is proportionally (2,ℓ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(2, \ell )$$\end{document}-choosable if and only if it is proportionally 2-choosable.
TI - Proportional 2-Choosability with a Bounded Palette
JF - Graphs and Combinatorics
DO - 10.1007/s00373-021-02448-w
DA - 2022-02-01
UR - https://www.deepdyve.com/lp/springer-journals/proportional-2-choosability-with-a-bounded-palette-RdOm0sII0p
VL - 38
IS - 1
DP - DeepDyve
ER -