Generalized Colorings of Graphs

A graph coloring is an assignment of labels called “colors” to certain elements of a graph subject to certain constraints. The proper vertex coloring is the most common type of graph coloring, where each vertex of a graph is assigned one color such that no two adjacent vertices share the same color, with the objective of minimizing the number of colors used. One can obtain various generalizations of the proper vertex coloring problem, by strengthening or relaxing the constraints or changing the objective. We study several types of such generalizations in this thesis. Series-parallel graphs are multigraphs that have no K4-minor. We provide bounds on their fractional and circular chromatic numbers and the defective version of these pa-rameters. In particular we show that the fractional chromatic number of any series-parallel graph of odd girth k is exactly 2k/(k − 1), confirming a conjecture by Wang and Yu. We introduce a generalization of defective coloring: each vertex of a graph is assigned a fraction of each color, with the total amount of colors at each vertex summing to 1. We define the fractional defect of a vertex v to be the sum of the overlaps with each neighbor of v, and the fractional defect of the graph to be the maximum of the defects over all vertices. We provide results on the minimum fractional defect of 2-colorings of some graphs. We also propose some open questions and conjectures. Given a (not necessarily proper) vertex coloring of a graph, a subgraph is called rainbow if all its vertices receive different colors, and monochromatic if all its vertices receive the same color. We consider several types of coloring here: a no-rainbow-F coloring of G is a coloring of the vertices of G without rainbow subgraph isomorphic to F ; an F -WORM coloring of G is a coloring of the vertices of G without rainbow or monochromatic subgraph isomorphic to F ; an (M, R)-WORM coloring of G is a coloring of the vertices of G with neither a monochromatic subgraph isomorphic to M nor a rainbow subgraph isomorphic to R. We present some results on these concepts especially with regards to the existence of colorings, complexity, and optimization within certain graph classes. Our focus is on the case that F , M or R is a path, cycle, star, or clique

Advances in Discrete Applied Mathematics and Graph Theory

The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs

Optimization Methods for Cluster Analysis in Network-based Data Mining

This dissertation focuses on two optimization problems that arise in network-based data mining, concerning identification of basic community structures (clusters) in graphs: the maximum edge weight clique and maximum induced cluster subgraph problems. We propose a continuous quadratic formulation for the maximum edge weight clique problem, and establish the correspondence between its local optima and maximal cliques in the graph. Subsequently, we present a combinatorial branch-and-bound algorithm for this problem that takes advantage of a polynomial-time solvable nonconvex relaxation of the proposed formulation. We also introduce a linear-time-computable analytic upper bound on the clique number of a graph, as well as a new method of upper-bounding the maximum edge weight clique problem, which leads to another exact algorithm for this problem. For the maximum induced cluster subgraph problem, we present the results of a comprehensive polyhedral analysis. We derive several families of facet-defining valid inequalities for the IUC polytope associated with a graph. We also provide a complete description of this polytope for some special classes of graphs. We establish computational complexity of the separation problems for most of the considered families of valid inequalities, and explore the effectiveness of employing the corresponding cutting planes in an integer (linear) programming framework for the maximum induced cluster subgraph problem

THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a

Graph Theory

Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem

Modeling and Tuning of Energy Harvesting Device Using Piezoelectric Cantilever Array

Piezoelectric devices have been increasingly investigated as a means of converting ambient vibrations into electrical energy that can be stored and used to power other devices, such as the sensors/actuators, micro-electro-mechanical systems (MEMS) devices, and microprocessor units etc. The objective of this work was to design, fabricate, and test a piezoelectric device to harvest as much power as possible from vibration sources and effectively store the power in a battery.;The main factors determining the amount of collectable power of a single piezoelectric cantilever are its resonant frequency, operation mode and resistive load in the charging circuit. A proof mass was used to adjust the resonant frequency and operation mode of a piezoelectric cantilever by moving the mass along the cantilever. Due to the tiny amount of collected power, a capacitor was suggested in the charging circuit as an intermediate station. To harvest sufficient energy, a piezoelectric cantilever array, which integrates multiple cantilevers in parallel connection, was investigated.;In the past, most prior research has focused on the theoretical analysis of power generation instead of storing generated power in a physical device. In this research, a commercial solid-state battery was used to store the power collected by the proposed piezoelectric cantilever array. The time required to charge the battery up to 80% capacity using a constant power supply was 970 s. It took about 2400 s for the piezoelectric array to complete the same task. Other than harvesting energy from sinusoidal waveforms, a vibration source that emulates a real environment was also studied. In this research the response of a bridge-vehicle system was used as the vibration sources such a scenario is much closer to a real environment compared with typical lab setups

Roman Domination in Complementary Prism Graphs

A Roman domination function on a complementary prism graph GGc is a function f : V [ V c ! {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number R(GGc) of a graph G = (V,E) is the minimum of Px2V [V c f(x) over such functions, where the complementary prism GGc of G is graph obtained from disjoint union of G and its complement Gc by adding edges of a perfect matching between corresponding vertices of G and Gc. In this paper, we have investigated few properties of R(GGc) and its relation with other parameters are obtaine

Fractional refinements of integral theorems

The focus of this thesis is to take theorems which deal with ``integral" objects in graph theory and consider fractional refinements of them to gain additional structure. A classic theorem of Hakimi says that for an integer $k$, a graph has maximum average degree at most $2k$ if and only if the graph decomposes into $k$ pseudoforests. To find a fractional refinement of this theorem, one simply needs to consider the instances where the maximum average degree is fractional. We prove that for any positive integers $k$ and $d$, if $G$ has maximum average degree at most $2k + \frac{2d}{k+d+1}$, then $G$ decomposes into $k+1$ pseudoforests, where one of pseudoforests has every connected component containing at most $d$ edges, and further this pseudoforest is acyclic. The maximum average degree bound is best possible for every choice of $k$ and $d$. Similar to Hakimi's Theorem, a classical theorem of Nash-Williams says that a graph has fractional arborcity at most $k$ if and only if $G$ decomposes into $k$ forests. The Nine Dragon Tree Theorem, proven by Jiang and Yang, provides a fractional refinement of Nash-Williams Theorem. It says, for any positive integers $k$ and $d$, if a graph $G$ has fractional arboricity at most $k + \frac{d}{k+d+1}$, then $G$ decomposes into $k+1$ forests, where one of the forests has maximum degree $d$. We prove a strengthening of the Nine Dragon Tree Theorem in certain cases. Let $k=1$ and $d \in \{3,4\}$. Every graph with fractional arboricity at most $1 + \frac{d}{d+2}$ decomposes into two forests $T$ and $F$ where $F$ has maximum degree $d$, every component of $F$ contains at most one vertex of degree $d$, and if $d= 4$, then every component of $F$ contains at most $8$ edges $e=xy$ such that both $\deg(x) \geq 3$ and $\deg(y) \geq 3$. In fact, when $k = 1$ and $d=3$, we prove that every graph with fractional arboricity $1 + \frac{3}{5}$ decomposes into two forests $T,F$ such that $F$ has maximum degree $3$, every component of $F$ has at most one vertex of degree $3$, further if a component of $F$ has a vertex of degree $3$ then it has at most $14$ edges, and otherwise a component of $F$ has at most $13$ edges. Shifting focus to problems which partition the vertex set, circular colouring provides a way to fractionally refine colouring problems. A classic theorem of Tuza says that every graph with no cycles of length $1 \bmod k$ is $k$-colourable. Generalizing this to circular colouring, we get the following: Let $k$ and $d$ be relatively prime, with $k>2d$, and let $s$ be the element of $\mathbb{Z}_k$ such that $sd \equiv 1\mod k$. Let $xy$ be an edge in a graph $G$. If $G-xy$ is $(k,d)$-circular-colorable and $G$ is not, then $xy$ lies in at least one cycle in $G$ of length congruent to $is \mod k$ for some $i$ in $\{1,\ldots,d\}$. If this does not occur with $i \in\{1,\ldots,d-1\}$, then $xy$ lies in at least two cycles of length $1 \mod k$ and $G-xy$ contains a cycle of length $0 \mod k$. This theorem is best possible with regards to the number of congruence classes when $k = 2d+1$. A classic theorem of Gr\"{o}tzsch says that triangle free planar graphs are $3$-colourable. There are many generalizations of this result, however fitting the theme of fractional refinements, Jaeger conjectured that every planar graph of girth $4k$ admits a homomorphism to $C_{2k+1}$. While we make no progress on this conjecture directly, one way to approach the conjecture is to prove critical graphs have large average degree. On this front, we prove: Every $4$-critical graph which does not have a $(7,2)$-colouring and is not $K_{4}$ or $W_{5}$ satisfies $e(G) \geq \frac{17v(G)}{10}$, and every triangle free $4$-critical graph satisfies $e(G) \geq \frac{5v(G)+2}{3}$. In the case of the second theorem, a result of Davies shows there exists infinitely many triangle free $4$-critical graphs satisfying $e(G) = \frac{5v(G) +4}{3}$, and hence the second theorem is close to being tight. It also generalizes results of Thomas and Walls, and also Thomassen, that girth $5$ graphs embeddable on the torus, projective plane, or Klein bottle are $3$-colourable. Lastly, a theorem of Cereceda, Johnson, and van den Heuvel, says that given a $2$-connected bipartite planar graph $G$ with no separating four-cycles and a $3$-colouring $f$, then one can obtain all $3$-colourings from $f$ by changing one vertices' colour at a time if and only if $G$ has at most one face of size $6$. We give the natural generalization of this to circular colourings when $\frac{p}{q} < 4$