Coloring and covering problems on graphs
type: CreativeWork
description: The \emph{separation dimension} of a graph $G$, written $\pi(G)$, is the minimum number of linear orderings of $V(G)$ such that every two nonincident edges are ``separated'' in some ordering, meaning that both endpoints of one edge appear before both endpoints of the other. We introduce the \emph{fractional separation dimension} $\pi_f(G)$, which is the minimum of $a/b$ such that some $a$ linear orderings (repetition allowed) separate every two nonincident edges at least $b$ times. In contrast to separation dimension, we show fractional separation dimension is bounded: always $\pi_f(G)\le 3$, with equality if and only if $G$ contains $K_4$. There is no stronger bound even for bipartite graphs, since $\pi_f(K_{m,m})=\pi_f(K_{m+1,m})=\frac{3m}{m+1}$. We also compute $\pi_f(G)$ for cycles and some complete tripartite graphs. We show that $\pi_f(G)<\sqrt{2}$ when $G$ is a tree and present a sequence of trees on which the value tends to $4/3$. We conjecture that when $n=3m$ the $K_4$-free $n$-vertex graph maximizing $\pi_f(G)$ is $K_{m,m,m}$. We also consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let $\pi^\circ(G)$ be the number of circular orderings needed to separate all pairs, and let $\pi_f^\circ(G)$ be the fractional version. Among our results: (1) $\pi^\circ(G)=1$ if and only $G$ is outerplanar. (2) $\pi^\circ(G)\le2$ when $G$ is bipartite. (3) $\pi^\circ(K_n)\ge\log_2\log_3(n-1)$. (4) $\pi_f^\circ(G)\le\frac{3}{2}$, with equality if and only if $K_4\subseteq G$. (5) $\pi_f^\circ(K_{m,m})=\frac{3m-3}{2m-1}$. A \emph{star $k$-coloring} is a proper $k$-coloring where the union of any two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an \emph{\Ifp}. We use discharging to pro…
abstract: The \emph{separation dimension} of a graph $G$, written $\pi(G)$, is the minimum number of linear orderings of $V(G)$ such that every two nonincident edges are ``separated'' in some ordering, meaning that both endpoints of one edge appear before both endpoints of the other. We introduce the \emph{fractional separation dimension} $\pi_f(G)$, which is the minimum of $a/b$ such that some $a$ linear orderings (repetition allowed) separate every two nonincident edges at least $b$ times. In contrast to separation dimension, we show fractional separation dimension is bounded: always $\pi_f(G)\le 3$, with equality if and only if $G$ contains $K_4$. There is no stronger bound even for bipartite graphs, since $\pi_f(K_{m,m})=\pi_f(K_{m+1,m})=\frac{3m}{m+1}$. We also compute $\pi_f(G)$ for cycles and some complete tripartite graphs. We show that $\pi_f(G)<\sqrt{2}$ when $G$ is a tree and present a sequence of trees on which the value tends to $4/3$. We conjecture that when $n=3m$ the $K_4$-free $n$-vertex graph maximizing $\pi_f(G)$ is $K_{m,m,m}$. We also consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let $\pi^\circ(G)$ be the number of circular orderings needed to separate all pairs, and let $\pi_f^\circ(G)$ be the fractional version. Among our results: (1) $\pi^\circ(G)=1$ if and only $G$ is outerplanar. (2) $\pi^\circ(G)\le2$ when $G$ is bipartite. (3) $\pi^\circ(K_n)\ge\log_2\log_3(n-1)$. (4) $\pi_f^\circ(G)\le\frac{3}{2}$, with equality if and only if $K_4\subseteq G$. (5) $\pi_f^\circ(K_{m,m})=\frac{3m-3}{2m-1}$. A \emph{star $k$-coloring} is a proper $k$-coloring where the union of any two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an \emph{\Ifp}. We use discharging to pro…
author: Loeb, Sarah Jane
language: en
keywords: Graph coloring, Graph covering
date published: 2017-09-29T17:56:39Z
date modified: 2026-07-22T22:24:35Z
provider: University of Illinois - Urbana-Champaign
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Illinois at Urbana-Champaign

PropertyValue
property ID: Handle
value: http://hdl.handle.net/2142/98358