University of Macau / Faculty of Science and Technology / Department of Computer and Information Science

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Analyzes the results of M.I. Ostrovskii's theorem of inequalities which estimate the minimal edge congestion for finite simple graphs. Uses the generic results of the theorem to examine and further reduce the parameters of inequalities for specific families of graphs, particularly complete graphs and complete bipartite graphs. Also, explores a possible minimal congestion tree for some grids while forming a conjecture for all grids.

abstract: The chromatic number $\chi(G)$ of a graph $G=(V,E)$ is the minimum number of colors needed to color $V(G)$ such that no adjacent vertices receive the same color. The coloring number $\col(G)$ of a graph $G$ is the minimum number $k$ such that there exists a linear ordering of $V(G)$ for which each vertex has at most $k-1$ backward neighbors. It is well known that the coloring number is an upper bound for the chromatic number. The weak $r$-coloring number $\wcol_{r}(G)$ is a generalization of the coloring number, and it was first introduced by Kierstead and Yang \cite{77}. The weak $r$-coloring number $\wcol_{r}(G)$ is the minimum integer $k$ such that for some linear ordering $L$ of $V(G)$ each vertex $v$ can reach at most $k-1$ other smaller vertices $u$ (with respect to $L$) with a path of length at most $r$ and $u$ is the smallest vertex in the path. This dissertation proves that $\wcol_{2}(G)\le23$ for every planar graph $G$. The exact distance-$3$ graph $G^{[\natural3]}$ of a graph $G=(V,E)$ is a graph with $V$ as its set of vertices, and $xy\in E(G^{[\natural3]})$ if and only if the distance between $x$ and $y$ in $G$ is $3$. This dissertation improves the best known upper bound of the chromatic number of the exact distance-$3$ graphs $G^{[\natural3]}$ of planar graphs $G$, which is $105$, to $95$. It also improves the best known lower bound, which is $7$, to $9$. A class of graphs is nowhere dense if for every $r\ge 1$ there exists $t\ge 1$ such that no graph in the class contains a topological minor of the complete graph $K_t$ where every edge is subdivided at most $r$ times. This dissertation gives a new characterization of nowhere dense classes using generalized notions of the domination number. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2020

A $(m,n)$-mixed graph is a mixed graph whose edges are assigned one of $m$ colours, and whose arcs are assigned one of $n$ colours. Let $G$ be a $(m,n)$-mixed graph and $\pi=(\alpha,\beta,\gamma_1,\gamma_2,\ldots,\gamma_n)$ be a $(n+2)$-tuple of permutations from $S_m \times S_n \times S_2^n$. We define \emph{switching at a vertex $v$ with respect to $\pi$} as follows. Replace each edge $vw$ of colour $\phi$ by an edge $vw$ of colour $\alpha(\phi)$, and each arc $vx$ of colour $\phi$ by an arc $\gamma_\phi(vx)$ of colour $\beta(\phi)$. In this thesis, we study the complexity of the question: ``Given a $(m,n)$-mixed graph $G$, is there a sequence of switches at vertices of $G$ with respect to the fixed group $\Gamma$ so that the resulting $(m,n)$-mixed graph admits a homomorphism to some $(m,n)$-mixed graph on $2$ vertices?'' We show the following: (1) When restricted to $(m,0)$-mixed graphs $H$ on at most $2$ vertices, the $\Gamma$-switchable homomorphism decision problem is solvable in polynomial time; (2) for each bipartite $(0,n)$-mixed graph $H$, there is a bipartite $(2n,0)$-mixed graph such that the respective $\Gamma$-switchable homomorphism decision problems are polynomially equivalent; (3) For all $(m,n)$-mixed graphs and groups, when $H$ has at most $2$ vertices, the $\Gamma$-switchable homomorphism decision problem is polynomial time solvable; (4) For a yes-instance of the $\Gamma$-switchable homomorphism problem for $(m,0)$-mixed graphs, we can find in quadratic time a sequence of switches on $G$ such that the resulting $(m,0)$-mixed graph admits a homomorphism to $H$. By proving (1)-(4), we show that the $\Gamma$-switchable $2$-colouring problem for $(m,n)$-mixed graphs is solvable in polynomial time for all finite permutation groups $\Gamma$ and provide a step towards a dichotomy theorem for the complexity of the $\Gamma$-switchable homomorphism decision problem. / Graduate

Given a finite set V, a convexity, C, is a collection of subsets of V that contains both the empty set and the set V and is closed under intersections. The elements of C are called convex sets. We can define several different convexities on the vertex set of a graph. In particular, the digital convexity, originally proposed as a tool for processing digital images, is defined as follows: a subset S of V(G) is digitally convex if, for every vertex v in V(G), we have N[v] a subset of N[S] implies v in S. Or, in other words, each vertex v that is not in the digitally convex set S needs to have a private neighbour in the graph with respect to S. In this thesis, we focus on the generation and enumeration of digitally convex sets in several classes of graphs. We establish upper bounds on the number of digitally convex sets of 2-trees, k-trees and simple clique 2-trees, as well as conjecturing a lower bound on the number of digitally convex sets of 2-trees and a generalization to k-trees. For other classes of graphs, including powers of cycles and paths, and Cartesian products of complete graphs and of paths, we enumerate the digitally convex sets using recurrence relations. Finally, we enumerate the digitally convex sets of block graphs in terms of the number of blocks in the graph, rather than in terms of the order of the graph. / Graduate

The astrid is a family of graphs in the Cartesian plane that frequently appears as an enrichment activity in the elementary setting. Many variations of the astrid exist, but the most common is a finite collection of straight line segments. The exploration of the graph theoretic properties of this version of the astrid will be the main focus of this paper.

In this paper, we give necessary and sufficient conditions for the existence of a decomposition of the λ-fold mixed complete graph into partial orientations of P4 and S3. Simple direct constructions are given in each case.

Let v ∈ Z+ and G be a simple graph. A G-decomposition of Kv is a collection F={F1,F2,...,Ft} of subgraphs of Kv such that every edge of Kv occurs in exactlyone of the subgraphs and every graph Fi ∈ F is isomorphic to G. A G-decomposition of Kv is called balanced if each vertex of Kv occurs in the same number of copies of G. In 2011, Dukes and Malloch provided an existence theory for balanced G-decompositions of Kv. Shortly afterwards, Bonisoli, Bonvicini, and Rinaldi introduced degree- and orbit-balanced G-decompositions. Similar to balanced decompositions,these two types of G-decompositions impose a local structure on the vertices of Kv. In this thesis, we will present an existence theory for degree- and orbit-balanced G-decompositions of Kv. To do this, we will first develop a theory for decomposing Kv into copies of G when G contains coloured loops. This will be followed by a brief discussion about the applications of such decompositions. Finally, we will explore anextension of this problem where Kv is decomposed into a family of graphs. We will examine the complications that arise with families of graphs and provide results for a few special cases. / Graduate