Non-crossing shortest paths in planar graphs with applications to max flow, and path graphs

This thesis is concerned with non-crossing shortest paths in planar graphs with applications to st-max flow vitality and path graphs. In the first part we deal with non-crossing shortest paths in a plane graph G, i.e., a planar graph with a fixed planar embedding, whose extremal vertices lie on the same face of G. The first two results are the computation of the lengths of the non-crossing shortest paths knowing their union, and the computation of the union in the unweighted case. Both results require linear time and we use them to describe an efficient algorithm able to give an additive guaranteed approximation of edge and vertex vitalities with respect to the st-max flow in undirected planar graphs, that is the max flow decrease when the edge/vertex is removed from the graph. Indeed, it is well-known that the st-max flow in an undirected planar graph can be reduced to a problem of non-crossing shortest paths in the dual graph. We conclude this part by showing that the union of non-crossing shortest paths in a plane graph can be covered with four forests so that each path is contained in at least one forest. In the second part of the thesis we deal with path graphs and directed path graphs, where a (directed) path graph is the intersection graph of paths in a (directed) tree. We introduce a new characterization of path graphs that simplifies the existing ones in the literature. This characterization leads to a new list of local forbidden subgraphs of path graphs and to a new algorithm able to recognize path graphs and directed path graphs. This algorithm is more intuitive than the existing ones and does not require sophisticated data structures

Implementing path coloring algorithms on planar graphs

Master's Project (M.S.) University of Alaska Fairbanks, 2017A path coloring of a graph partitions its vertex set into color classes such that each class induces a disjoint union of paths. In this project we implement several algorithms to compute path colorings of graphs embedded in the plane. We present two algorithms to path color plane graphs with 3 colors based on a proof by Poh in 1990. First we describe a naive algorithm that directly follows Poh's procedure, then we give a modified algorithm that runs in linear time. Independent results of Hartman and Skrekovski describe a procedure that takes a plane graph G and a list of 3 colors for each vertex, and computes a path coloring of G such that each vertex receives a color from its list. We present a linear time implementation based on Hartman and Skrekovski's proofs. A C++ implementation is provided for all three algorithms, utilizing the Boost Graph Library. Instructions are given on how to use the implementation to construct colorings for plane graphs represented by Boost data structures

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

List Coloring Some Classes of 1-Planar Graphs

In list coloring we are given a graph G and a list assignment for G which assigns to each vertex of G a list of possible colors. We wish to find a coloring of the vertices of G such that each vertex uses a color from its list and adjacent vertices are given different colors. In this thesis we study the problem of list coloring 1-planar graphs, i.e., graphs that can be drawn in the plane such that any edge intersects at most one other edge. We also study the closely related problem of simultaneously list coloring the vertices and faces of a planar graph, known as coupled list coloring. We show that 1-planar bipartite graphs are list colorable whenever all lists are of size at least four, and further show that this coloring can be found in linear time. In pursuit of this result, we show that the previously known edge partition of a 1-planar graph into a planar graph and a forest can be found in linear time. A wheel graph consists of a cycle of vertices, all of which are adjacent to an additional center vertex. We show that wheel graphs are coupled list colorable when all lists are of size five or more and show that this coloring can be found in linear time. Possible extensions of this result to planar partial 3-trees are discussed. Finally, we discuss the complexity of list coloring 1-planar graphs, both in parameterized and unparameterized settings

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

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 $\frac{p}{q} < 4$

Positive Hopf plumbed links with maximal signature

This thesis has six chapters. In Chapters 1 and 2, we give the definitions and examples of the main links under study in this thesis: positive braids links, checkerboard graph links, and basket links. In Chapter 2, we also define the signature function of a link and compute it for the cases of three and four positive Hopf bands plumbed together. In particular, we investigate the behavior of the signature function when the intersection of the core curves of these Hopf bands goes to infinity. In Chapter 3, we define special types of moves for checkerboard graphs and use them to prove that a checkerboard graph link with maximal signature is isotopic to one of the links realized by the simply laced Dinkin diagrams (ADE diagrams). In Chapter 4, we also characterize checkerboard graphs whose corresponding link is of the ADE type and use this to show that there is a linear time algorithm to find these links from a checkerboard graph. In Chapter 5, we study the connection between checkerboard graph links and basket links. In Chapter 6, we prove that a basket link made with positive Hopf bands and symmetrized Seifert form congruent to the Cartan matrix of the simply laced Dynkin diagram A is isotopic to a two-strand torus link. We also provide some examples of fake A links