Extremal problems in graph theory

We consider generalized graph coloring and other extremal problems in graph theory. We also construct twisted hypercubes of small radius and find the domination number of the Kneser graph $K(n,k)$ when $n\ge{3\over4}k\sp2\pm k,$ depending on whether k is even or odd. The path chromatic number $\chi\sb{P}(G)$ of a graph G is the least number of colors with which the vertices of G can be colored so that each color class induces a disjoint union of paths. We characterize cartesian products of cycles with path chromatic number 2. We show that if G is a toroidal graph, then for any non-contractible chordless cycle C of G, there is a 3-coloring of the vertices of G so that each color class except one induces a disjoint union of paths, while the third color class induces a disjoint union of paths and the cycle C. The path list chromatic number of a graph, $\\chi\sb{P}(G),$ is the minimum k for which, given any assignment of lists of size k to each vertex, G can be colored by assigning each vertex a color from its list so that each color class induces a disjoint union of paths. We prove that $\\chi\sb{P}(G)\le3.$ The observability of a graph G is least number of colors in a proper edge-coloring of G such that the color sets at vertices of G are pairwise distinct. A graph G has a set-balanced k-edge-coloring if the edges of G can be properly colored with k colors so that, for each degree, the color sets at vertices of that degree occur with multiplicities differing by at most one. We determine the values of k such that G has a set-balanced k-edge-coloring whenever G is a member of various classes of graphs. The spot-chromatic number of a graph, $\chi\sb{S}(G),$ is the least number of colors with which the vertices of G can be colored so that each color class induces a disjoint union of cliques. We show that \chi\sb{S}(K\sb{mt} \square K\sb{nt})\le{mnt\over m+n}+2\min(m,n) whenever $m+n$ divides t. Let ${\cal G}\sb0=\{K\sb1\}.$ For $k\ge1,$ the family ${\cal G}\sb{k}$ of twisted hypercubes of dimension k is the set of graphs constructible by adding a matching joining two graphs in ${\cal G}\sb{k-1}.$ We construct a family of twisted hypercubes of small diameter. We prove that the order of growth of the minimum diameter among twisted hypercubes of dimension k is $\Theta(k$/lg k). The domination number γ(G) of a graph G is the minimum size of a set S such that every vertex of G is in S or is adjacent to some vertex in S. The Kneser graph $K(n, k)$ has as vertices the k-subsets of $\lbrack n\rbrack.$ We determine γ(K(n,k)) when $n\ge{3\over4}k\sp2\pm k$ depending on the parity of k.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2012

Author dc:creator

• Hartman, Christopher M.

Contributors dc:contributor

• West, Douglas B.
• Weichsel, Paul M.
• Francis, George K.
• Edelsbrunner, Herbert

• graph theory

Statement dc:rights

• Copyright 1997 Christopher M. Hartman

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/29956

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/29956