In this thesis we give two different proofs of the result chromatic number of a special graph is 4. The first proof is derived by analysing the structure of the special graph. The second proof is a method which was first studied in [1].

This thesis constructs special graphs with given circular chromatic numbers or circular flow numbers. Suppose $G=(V,E)$ is a graph and $rgeq 2$ is a real number. An $r$-coloring of a graph $G$ is a mapping $f:V ightarrow [0,r)$ such that for any adjacent vertices $x,y$ of $G$, $1leq |f(x)-f(y)|leq r-1$. The circular chromatic number $chi_c(G)$ is the least $r$ for which there exists an $r$-coloring of $G$. The circular chromatic number was introduced by Vince in 1988 in cite{vince}, where the parameter is called the {em star chromatic number} and denoted by $chi^*(G)$. Vince proved that for any rational number $k/dgeq 2$ there is a graph $G$ with $chi_c(G)=k/d$. In this thesis, we are interested in the existence of special graphs with given circular chromatic numbers. A graph $H$ is called a minor of a graph $G$ if $H$ can be obtained from $G$ by deleting some vertices and edges, and contracting some edges. A graph $G$ is called $H$-minor free if $H$ is not a minor of G. The well-known Hadwiger's conjecture asserts that for any positive integer $n$, any $K_n$-minor free graph $G$ is $(n-1)$-colorable. If this conjecture is true, then for any $K_n$-minor free graph $G$, we have $chi_c(G)leq n-1$. On the other hand, for any graph $G$ with at least one edge we have $chi_c(G)geq 2$. A natural question is this: Is it true that for any rational number $2leq rleq n-1$, there exist a $K_n$-minor free graph $G$ with $chi_c(G)=r$? For $n=4$, the answer is ``no". It was proved by Hell and Zhu in cite{hz98} that if $G$ is a $K_4$-minor free graph then either $chi_c(G)=3$ or $chi_c(G)leq 8/3$. So none of the rational numbers in the interval $(8/3,3)$ is the circular chromatic number of a $K_4$-minor free graph. For $ngeq 5$, Zhu cite{survey} proved that for any rational number $rin[2,n-2]$, there exists a $K_n$-minor free graph $G$ with $chi_c(G)=r$. The question whether there exists a $K_n$-minor free graph $G$ with $chi_c(G)=r$ for each rational number $rin(n-2,n-1)$ remained open. In this thesis, we answer this question in the affirmative. For each integer $ngeq 5$, for each rational number $rin[n-2,n-1]$, we construct a $K_n$-minor free graph $G$ with $chi_c(G)=r$. This implies that for each $ngeq 5$, for each rational number $rin[2,n-1]$, there exists a $K_n$-minor free graph $G$ with $chi_c(G)=r$. In case $n=5$, the $K_5$-minor free graphs constructed in this thesis are actually planar graphs. So our result implies that for each rational number $rin[2,4]$, there exists a planar graph $G$ with $chi_c(G)=r$. This result was first proved by Moser cite{moser} and Zhu cite{3-4}. To be precise, Moser cite{moser} proved that for each rational number $rin[2,3]$, there exist a planar graph $G$ with $chi_c(G)=r$, and Zhu cite{3-4} proved that for each rational number $rin[3,4]$, there exists a planar graph $G$ with $chi_c(G)=r$. Moser's and Zhu's proofs are quite complicated. Our construction is conceptually simpler. Moreover, for $ngeq 5$, $K_n$-minor free graphs, including the planar graphs are constructed with a unified method. For $K_4$-minor free graphs, although Hell and Zhu cite{hz98} proved that there is no $K_4$-minor free graph $G$ with $chi_c(G)in (8/3,3)$. The question whether there exists a $K_4$-minor free graph $G$ with $chi_c(G)=r$ for each rational number $rin[2,8/3]$ remained open. This thesis solves this problem: For each rational number $rin[2,8/3]$, we shall construct a $K_4$-minor free $G$ with $chi_c(G)=r$. This thesis also studies the relation between the circular chromatic number and the girth of $K_4$-minor free graphs. For each integer $n$, the supremum of the circular chromatic number of $K_4$-minor free graphs of odd girth (the length of shortest odd cycle) at least $n$ is determined. It is also proved that the same bound is sharp for $K_4$-minor free graphs of girth $n$. By a classical result of ErdH{o}s, for any positive integers $l$ and $n$, there exists a graph $G$ of girth at least $l$ and of chromatic number $n$. Using probabilistic method, Zhu cite{unique} proved that for each integer $l$ and each rational number $rgeq 2$, there is a graph $G$ of girth at least $l$ such that $chi_c(G)=r$. Construction of such graphs for $rgeq 3$ was given by Nev{s}etv{r}il and Zhu cite{nz}. The question of how to construct large girth graph $G$ with $chi_c(G)=r$ for given $rin(2,3)$ remained open. In this thesis, we present a unified method that constructs, for any $rgeq 2$, a graph $G$ of girth at least $l$ with circular chromatic number $chi_c(G) =r$. Graphs $G$ with $chi_c(G)=chi(G)$ have been studied extensively in the literature. Many families of graphs $G$ are known to satisfy $chi_c(G)=chi(G)$. However it remained as an open question as how to construct arbitrarily large $chi$-critical graphs $G$ of bounded maximum degree with $chi_c(G)=chi(G)$. This thesis presents a construction of such graphs. The circular flow number $Phi_c(G)$ is the dual concept of $chi_c(G)$. Let $G$ be a graph. Replace each edge $e=xy$ by a pair of opposite arcs $a=overrightarrow{xy}$ and $a^{-1}=overrightarrow{yx}$. We obtain a symmetric directed graph. Denote by $A(G)$ the set of all arcs of $G$. A chain is a mapping $f:A(G) ightarrow I!!R$ such that for each arc $a$, $f(a^{-1})=-f(a)$. A flow is a chain such that for each subset $X$ of $V(G)$, $sum_{ain[X,ar{X}]}f(a)=0$, where $[X,ar{X}]$ is the set of all arcs from $X$ to $V-X$. An $r$-flow is a flow such that for any arc $ain A(G)$ , $1leq |f(a)| leq r-1$. The circular flow number of $G$ is $Phi_c(G)=mbox{ inf}{r: G mbox{ admits a } rmbox{-flow}}$. It was conjectured by Tutte that every graph $G$ has $Phi_c(G)leq 5$. By taking the geometrical dual of planar graphs, Moser's and Zhu's results concerning circular chromatic numbers of planar graphs imply that for each rational number $rin[2,4]$, there is a graph $G$ with $Phi_c(G)=r$. The question remained open whether for each $rin(4,5)$, there exists a graph $G$ with $Phi_c(G)=r$. In this thesis, for each rational number $rin [4,5]$, we construct a graph $G$ with $Phi_c(G)=r$.

In (J Graph Theory 33 (2000), 14-24), Hell and Zhu proved that if a series-parallel graph G has girth at least 2⌊(3k-1)/2⌋, then χc(G)≤4k/(2k-1). In (J Graph Theory 33 (2000), 185-198), Chien and Zhu proved that the girth condition given in (J Graph Theory 33 (2000), 14-24) is sharp. Short proofs of both results are given in this note.

This dissertation has three principal components. The first component is about the connections between the dimension of posets and the size of matchings in comparability and incomparability graphs. In 1951, Hiraguchi proved that for any finite poset P, the dimension of P is at most half of the number of points in P. We develop some new inequalities for the dimension of finite posets. These inequalities are then used to bound dimension in terms of the maximum size of matchings. We prove that if the dimension of P is d and d is at least 3, then there is a matching of size d in the comparability graph of P, and a matching of size d in the incomparability graph of P. The bounds in above theorems are best possible, and either result has Hiraguchi's theorem as an immediate corollary. In the second component, we focus on an extremal graph theory problem whose solution relied on the construction of a special kind of posets. In 1959, Paul Erdos, in a landmark paper, proved the existence of graphs with arbitrarily large girth and arbitrarily large chromatic number using probabilistic method. In a 1991 paper of Kriz and Nesetril, they introduced a new graph parameter eye(G). They show that there are graphs with large girth and large chromatic number among the class of graphs having eye parameter at most three. Answering a question of Kriz and Nesetril, we were able to strengthen their results and show that there are graphs with large girth and large chromatic number among the class of graphs having eye parameter at most two. The last component is about random posets--the poset version of the Erdos-Renyi random graphs. In 1991, Erdos, Kierstead and Trotter (EKT) investigated random height 2 posets and obtained several upper and lower bounds on the dimension of the random posets. Motivated by some extremal problems involving conditions which force a poset to contain a large standard example, we were compelled to revisit this subject. Our sharpened analysis allows us to conclude that as p approaches 1, the expected value of dimension first increases and then decreases, a subtlety not identified in EKT. Along the way, we establish connections with classical topics in analysis as well as with latin rectangles. Also, using structural insights drawn from this research, we are able to make progress on the motivating extremal problem with an application of the asymmetric form of the Lovasz Local Lemma.

The generalized deBruijn graph dB(a, k) is the directed graph with a k vertices and edges between vertices x = a1, a 2, ... ak and y = b1, b2, ... b k precisely when a2, ... ak = b1, b2, ... bk-1. The deBruijn graphs can be further generalized by introducing an overlap variable t ≤ k - 1 where the number of consecutive digits by which the vertex labels (sequences) overlap is t. The α-overlap graph is the underlying simple graph of the generalized deBruijn digraph with vertex label overlap 0 < t ≤ k - 1.We denote the α-overlap graph by Gα = G(a, k, t) and the parameters a, k and t are positive integers such that a ≥ 2 and k > t > 0. Thus dB(a, k) = G(a, k, k - 1). In this paper, we show that every a-overlap graph is 3-colorable for any a if k is sufficiently large. We also determine bounds on the chromatic number of the α-overlap graphs if a is much larger than k.

No description available.

We study graph parameters arising from different types of colourings of random graphs, defined broadly as an assignment of colours to either the vertices or the edges of a graph. The chromatic number X(G) of a graph is the minimum number of colours required for a vertex colouring where no two adjacent vertices are coloured the same. Determining the chromatic number is one of the classic challenges in random graph theory. In Chapter 3, we give new upper and lower bounds for the chromatic number of the dense random graph G(n,p)) where p &isin; (0,1) is constant. These bounds are the first to match up to an additive term of order o(1) in the denominator, and in particular, they determine the average colour class size in an optimal colouring up to an additive term of order o(1). In Chapter 4, we study a related graph parameter called the equitable chromatic number. This is defined as the minimum number of colours needed for a vertex colouring where no two adjacent vertices are coloured the same and, additionally, all colour classes are as equal in size as possible. We prove one point concentration of the equitable chromatic number of the dense random graph G(n,m) with m = pn(n-1)/2, p &LT; 1-1/e² constant, on a subsequence of the integers. We also show that whp, the dense random graph G(n,p) allows an almost equitable colouring with a near optimal number of colours. We call an edge colouring of a graph G a rainbow colouring if every pair of vertices is joined by a rainbow path, which is a path where no colour is repeated. The least number of colours where this is possible is called the rainbow connection number rc(G). Since its introduction in 2008 as a new way to quantify how well connected a given graph is, the rainbow connection number has attracted the attention of a great number of researchers. For any graph G, rc(G)&ge;diam(G), where diam(G) denotes the diameter. In Chapter 5, we will see that in the random graph G(n,p), rainbow connection number 2 is essentially equivalent to diameter 2. More specifically, we consider G ~ G(n,p) close to the diameter 2 threshold and show that whp rc(G) = diam(G) &isin; {2,3}. Furthermore, we show that in the random graph process, whp the hitting times of diameter 2 and of rainbow connection number 2 coincide. In Chapter 6, we investigate sharp thresholds for the property rc(G)&le;=r where r is a fixed integer. The results of Chapter 6 imply that for r=2, the properties rc(G)&le;=2 and diam(G)&le;=2 share the same sharp threshold. For r&ge;3, the situation seems quite different. We propose an alternative threshold and prove that this is an upper bound for the sharp threshold for rc(G)&le;=r where r&ge;3.

A coloring of a graph is an assignment of colors to its vertices so that no two adjacent vertices are given the same color. The chromatic number of a graph is the least number of colors needed to color all of its vertices. Graph coloring problems can be applied to many real world applications, such as scheduling and register allocation. Computationally, the decision problem of whether a general graph is m-colorable is NP-complete for m ≥ 3. The graph studied in this thesis is a well-known combinatorial object, the k-dimensional hypercube, Qk. The hypercube itself is 2-colorable for all k; however, coloring the square of the cube is a much more interesting problem. This is the graph in which the vertices are binary vectors of length k, and two vertices are adjacent if and only if the Hamming distance between the two vectors is at most 2. Any color class in a coloring of Q2k is a binary (k;M, 3) code. This thesis will begin with an introduction to binary codes and their structure. One of the most fundamental combinatorial problems is finding optimal binary codes, that is, binary codes with the maximum cardinality satisfying a specified length and minimum distance. Many upper and lower bounds have been produced, and we will analyze and apply several of these. This leads to many interesting results about the chromatic number of the square of the cube. The smallest k for which the chromatic number of Q2k is unknown is k = 8; however, it can be determined that this value is either 13 or 14. Computational approaches to determine the chromatic number of Q28 were performed. We were unable to determine whether 13 or 14 is the true value; however, much valuable insight was learned about the structure of this graph and the computational difficulty that lies within. Since a 13-coloring of Q28 must have between 9 and 12 color classes being (8; 20; 3) binary codes, this led to a thorough investigation of the structure of such binary codes.

A coloring of a graph is an assignment of colors to its vertices so that no two adjacent vertices are given the same color. The chromatic number of a graph is the least number of colors needed to color all of its vertices. Graph coloring problems can be applied to many real world applications, such as scheduling and register allocation. Computationally, the decision problem of whether a general graph is m-colorable is NP-complete for m ≥ 3. The graph studied in this thesis is a well-known combinatorial object, the k-dimensional hypercube, Qk. The hypercube itself is 2-colorable for all k; however, coloring the square of the cube is a much more interesting problem. This is the graph in which the vertices are binary vectors of length k, and two vertices are adjacent if and only if the Hamming distance between the two vectors is at most 2. Any color class in a coloring of Q2k is a binary (k;M, 3) code. This thesis will begin with an introduction to binary codes and their structure. One of the most fundamental combinatorial problems is finding optimal binary codes, that is, binary codes with the maximum cardinality satisfying a specified length and minimum distance. Many upper and lower bounds have been produced, and we will analyze and apply several of these. This leads to many interesting results about the chromatic number of the square of the cube. The smallest k for which the chromatic number of Q2k is unknown is k = 8; however, it can be determined that this value is either 13 or 14. Computational approaches to determine the chromatic number of Q28 were performed. We were unable to determine whether 13 or 14 is the true value; however, much valuable insight was learned about the structure of this graph and the computational difficulty that lies within. Since a 13-coloring of Q28 must have between 9 and 12 color classes being (8; 20; 3) binary codes, this led to a thorough investigation of the structure of such binary codes.

A coloring of a graph is an assignment of colors to its vertices so that no two adjacent vertices are given the same color. The chromatic number of a graph is the least number of colors needed to color all of its vertices. Graph coloring problems can be applied to many real world applications, such as scheduling and register allocation. Computationally, the decision problem of whether a general graph is m-colorable is NP-complete for m ≥ 3. The graph studied in this thesis is a well-known combinatorial object, the k-dimensional hypercube, Qk. The hypercube itself is 2-colorable for all k; however, coloring the square of the cube is a much more interesting problem. This is the graph in which the vertices are binary vectors of length k, and two vertices are adjacent if and only if the Hamming distance between the two vectors is at most 2. Any color class in a coloring of Q2k is a binary (k;M, 3) code. This thesis will begin with an introduction to binary codes and their structure. One of the most fundamental combinatorial problems is finding optimal binary codes, that is, binary codes with the maximum cardinality satisfying a specified length and minimum distance. Many upper and lower bounds have been produced, and we will analyze and apply several of these. This leads to many interesting results about the chromatic number of the square of the cube. The smallest k for which the chromatic number of Q2k is unknown is k = 8; however, it can be determined that this value is either 13 or 14. Computational approaches to determine the chromatic number of Q28 were performed. We were unable to determine whether 13 or 14 is the true value; however, much valuable insight was learned about the structure of this graph and the computational difficulty that lies within. Since a 13-coloring of Q28 must have between 9 and 12 color classes being (8; 20; 3) binary codes, this led to a thorough investigation of the structure of such binary codes. / Graduate Studies, College of (Okanagan) / Graduate