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 Title
 ON SATURATION NUMBERS OF RAMSEYMINIMAL GRAPHS.
 Creator

Davenport, Hunter M, Song, ZiXia, University of Central Florida
 Abstract / Description

Dating back to the 1930's, Ramsey theory still intrigues many who study combinatorics. Roughly put, it makes the profound assertion that complete disorder is impossible. One view of this problem is in edgecolorings of complete graphs. For forbidden graphs H1,...,Hk and a graph G, we write G "arrows" (H1,...,Hk) if every kedgecoloring of G contains a monochromatic copy of Hi in color i for some i=1,2,...,k. If c is a (red, blue)edgecoloring of G, we say c is a bad coloring if G contains...
Show moreDating back to the 1930's, Ramsey theory still intrigues many who study combinatorics. Roughly put, it makes the profound assertion that complete disorder is impossible. One view of this problem is in edgecolorings of complete graphs. For forbidden graphs H1,...,Hk and a graph G, we write G "arrows" (H1,...,Hk) if every kedgecoloring of G contains a monochromatic copy of Hi in color i for some i=1,2,...,k. If c is a (red, blue)edgecoloring of G, we say c is a bad coloring if G contains no red K3or blue K1,t under c. A graph G is (H1,...,Hk)Ramseyminimal if G arrows (H1,...,Hk) but no proper subgraph of G has this property. Given a family F of graphs, we say that a graph G is Fsaturated if no member of F is a subgraph of G, but for any edge xy not in E(G), G + xy contains a member of F as a subgraph. Letting Rmin(K3, K1,t) be the family of (K3,K1,t)Ramsey minimal graphs, we study the saturation number, denoted sat(n,Rmin(K3,K1,t)), which is the minimum number of edges among all Rmin(K3,K1,t)saturated graphs on n vertices. We believe the methods and constructions developed in this thesis will be useful in studying the saturation numbers of (K4,K1,t)saturated graphs.
Show less  Date Issued
 2018
 Identifier
 CFH2000291, ucf:45881
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFH2000291
 Title
 Interval EdgeColorings of Graphs.
 Creator

Foster, Austin, Song, Zixia, Reid, Michael, Brennan, Joseph, University of Central Florida
 Abstract / Description

A proper edgecoloring of a graph G by positive integers is called an interval edgecoloring if the colors assigned to the edges incident to any vertex in G are consecutive (i.e., those colors form an interval of integers). The notion of interval edgecolorings was first introduced by Asratian and Kamalian in 1987, motivated by the problem of finding compact school timetables. In 1992, Hansen described another scenario using interval edgecolorings to schedule parentteacher conferences so...
Show moreA proper edgecoloring of a graph G by positive integers is called an interval edgecoloring if the colors assigned to the edges incident to any vertex in G are consecutive (i.e., those colors form an interval of integers). The notion of interval edgecolorings was first introduced by Asratian and Kamalian in 1987, motivated by the problem of finding compact school timetables. In 1992, Hansen described another scenario using interval edgecolorings to schedule parentteacher conferences so that every person's conferences occur in consecutive slots. A solution exists if and only if the bipartite graph with vertices for parents and teachers, and edges for the required meetings, has an interval edgecoloring.A wellknown result of Vizing states that for any simple graph $G$, $\chi'(G) \leq \Delta(G) + 1$, where $\chi'(G)$ and $\Delta(G)$ denote the edgechromatic number and maximum degree of $G$, respectively. A graph $G$ is called class 1 if $\chi'(G) = \Delta(G)$, and class 2 if $\chi'(G) = \Delta(G) + 1$. One can see that any graph admitting an interval edgecoloring must be of class 1, and thus every graph of class 2 does not have such a coloring.Finding an interval edgecoloring of a given graph is hard. In fact, it has been shown that determining whether a bipartite graph has an interval edgecoloring is NPcomplete. In this thesis, we survey known results on interval edgecolorings of graphs, with a focus on the progress of $(a, b)$biregular bipartite graphs. Discussion of related topics and future work is included at the end. We also give a new proof of Theorem 3.15 on the existence of proper path factors of $(3, 4)$biregular graphs. Finally, we obtain a new result, Theorem 3.18, which states that if a proper path factor of any $(3, 4)$biregular graph has no path of length 8, then it contains paths of length 6 only. The new result we obtained and the method we developed in the proof of Theorem 3.15 might be helpful in attacking the open problems mentioned in the Future Work section of Chapter 5.
Show less  Date Issued
 2016
 Identifier
 CFE0006301, ucf:51609
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0006301