Thin Trees in Some Families of Graphs

Let =(,) be a graph and let be a spanning tree of . The thinness parameter of denoted by () is the maximum over all cuts of the proportion of the edges of in the cut. Thin trees play an important role in some recent papers on the Asymmetric Traveling Salesman Problem (ATSP). Goddyn conjectured that every graph of sufficiently large edge-connectivity has a spanning tree such that () ≤ . In this thesis, we study the problem of finding thin spanning trees in two families of graphs, namely, (1) distance-regular graphs (DRGs), and (2) planar graphs. For some families of DRGs such as strongly regular graphs, Johnson graphs, Crown graphs, and Hamming graphs, we give a polynomial-time construction of spanning trees of maximum degree ≤ 3 such that () is determined by the parameters of the graph. For planar graphs, we improve the analysis of Merker and Postle ("Bounded Diameter Arboricity", arXiv:1608.05352v1) and show that every 6-edge-connected planar graph has two edge-disjoint spanning trees ,′ such that (),(′) ≤ 14⁄15. For 8-edge-connected planar graphs , we present a simplified version of the techniques of Merker and Postle and show that has two edge-disjoint spanning trees ,′ such that (),(′) ≤ 12⁄13

Embedding oriented graphs in books

A book consists of a line L in [special characters omitted]3, called the spine, and a collection of half planes, called pages, whose common boundary is L. A k-book is book with k pages. A k-page book embedding is a continuous one-to-one mapping of a graph G into a book such that the vertices are mapped into L and the edges are each mapped to either the spine or a particular page, such that no two edges cross in any page. Each page contains a planar subgraph of G. The book thickness, denoted bt(G), is the minimum number of pages for a graph to have a k-page book embedding. We focus on oriented graphs, and propose a new way to embed oriented graphs into books, called an oriented book embedding, and define oriented book thickness . We investigate oriented graphs having oriented book thickness k using k-page critical oriented graphs, oriented graphs with oriented book thickness equal to k, but, for each arc, the deletion of that arc yields an oriented graph with oriented book thickness equal to k –1. We discuss several classes of two-page critical oriented graphs, and use them to characterize oriented graphs with oriented book thickness equal to one that are strictly uni-dicyclic graphs, oriented graphs having exactly one cycle, which is a directed cycle. We give a similar result for strictly bi-dicyclic graphs, oriented graphs having exactly two cycles, which are directed cycles

A study on Dicycles and Eulerian Subdigraphs

1. Dicycle cover of Hamiltonian oriented graphs. A dicycle cover of a digraph D is a family F of dicycles of D such that each arc of D lies in at least one dicycle in F. We investigate the problem of determining the upper bounds for the minimum number of dicycles which cover all arcs in a strong digraph. Best possible upper bounds of dicycle covers are obtained in a number of classes of digraphs, including strong tournaments, Hamiltonian oriented graphs, Hamiltonian oriented complete bipartite graphs, and families of possibly non-hamiltonian digraphs obtained from these digraphs via a sequence of 2-sum operations.;2. Supereulerian digraphs with given local structures . Catlin in 1988 indicated that there exist graph families F such that if every edge e in a graph G lies in a subgraph He of G isomorphic to a member in F, then G is supereulerian. In particular, if every edge of a connected graph G lies in a 3-cycle, then G is supereulerian. The purpose of this research is to investigate how Catlin\u27s theorem can be extended to digraphs. A strong digraph D is supereulerian if D contains a spanning eulerian subdigraph. We show that there exists an infinite family of non-supereulerian strong digraphs each arc of which lies in a directed 3-cycle. We also show that there exist digraph families H such that a strong digraph D is supereulerian if every arc a of D lies in a subdigraph Ha isomorphic to a member of H. A digraph D is symmetric if (x, y) ∈ A( D) implies (y, x) ∈ A( D); and is symmetrically connected if every pair of vertices of D are joined by a symmetric dipath. A digraph D is partially symmetric if the digraph obtained from D by contracting all symmetrically connected components is symmetrically connected. It is known that a partially symmetric digraph may not be symmetrically connected. We show that symmetrically connected digraphs and partially symmetric digraphs are such families. Sharpness of these results are discussed.;3. On a class of supereulerian digraphs. The 2-sum of two digraphs D1 and D2, denoted D1 ⊕2 D2, is the digraph obtained from the disjoint union of D 1 and D2 by identifying an arc in D1 with an arc in D2. A digraph D is supereulerian if D contains a spanning eulerian subdigraph. It has been noted that the 2-sum of two supereulerian (or even hamiltonian) digraphs may not be supereulerian. We obtain several sufficient conditions on D1 and D 2 for D1 ⊕2 D 2 to be supereulerian. In particular, we show that if D 1 and D2 are symmetrically connected or partially symmetric, then D1 ⊕2 D2 is supereulerian

Homomorphisms of (j,k)-mixed graphs

A mixed graph is a simple graph in which a subset of the edges have been assigned directions to form arcs. For non-negative integers j and k, a (j,k)−mixed graph is a mixed graph with j types of arcs and k types of edges. The collection of (j,k)−mixed graphs contains simple graphs ((0,1)−mixed graphs), oriented graphs ((1,0)−mixed graphs) and k−edge- coloured graphs ((0,k)−mixed graphs).A homomorphism is a vertex mapping from one (j,k)−mixed graph to another in which edge type is preserved, and arc type and direction are preserved. The (j,k)−chromatic number of a (j,k)−mixed graph is the least m such that an m−colouring exists. When (j,k)=(0,1), we see that these definitions are consistent with the usual definitions of graph homomorphism and graph colouring.In this thesis we study the (j,k)−chromatic number and related parameters for different families of graphs, focussing particularly on the (1,0)−chromatic number, more commonly called the oriented chromatic number, and the (0,k)−chromatic number.In addition to considering vertex colourings, we also consider incidence colourings of both graphs and digraphs. Using systems of distinct representatives, we provide a new characterisation of the incidence chromatic number. We define the oriented incidence chromatic number and find, by way of digraph homomorphism, a connection between the oriented incidence chromatic number and the chromatic number of the underlying graph. This connection motivates our study of the oriented incidence chromatic number of symmetric complete digraphs.Un graphe mixte est un graphe simple tel que un sous-ensemble des arêtes a une orientation. Pour entiers non négatifs j et k, un graphe mixte-(j,k) est un graphe mixte avec j types des arcs and k types des arêtes. La famille de graphes mixte-(j,k) contient graphes simple, (graphes mixte−(0,1)), graphes orienté (graphes mixte−(1,0)) and graphe coloré arête −k (graphes mixte−(0,k)).Un homomorphisme est un application sommet entre graphes mixte−(j,k) que tel les types des arêtes sont conservés et les types des arcs et leurs directions sont conservés. Le nombre chromatique−(j,k) d’un graphe mixte−(j,k) est le moins entier m tel qu’il existe un homomorphisme à une cible avec m sommets. Quand on observe le cas de (j,k) = (0,1), on peut déterminer ces définitions correspondent à les définitions usuel pour les graphes.Dans ce mémoire on etude le nombre chromatique−(j,k) et des paramètres similaires pour diverses familles des graphes. Aussi on etude les coloration incidence pour graphes and digraphs. On utilise systèmes de représentants distincts et donne une nouvelle caractérisation du nombre chromatique incidence. On define le nombre chromatique incidence orienté et trouves un connexion entre le nombre chromatique incidence orienté et le nombre chromatic du graphe sous-jacent

Oriented Colourings of Graphs with Maximum Degree Three and Four

We show that any orientation of a graph with maximum degree three has an oriented 9-colouring, and that any orientation of a graph with maximum degree four has an oriented 69-colouring. These results improve the best known upper bounds of 11 and 80, respectively

A study on supereulerian digraphs and spanning trails in digraphs

A strong digraph D is eulerian if for any v ∈ V (D), d+D (v) = d−D (v). A digraph D is supereulerian if D contains a spanning eulerian subdigraph, or equivalently, a spanning closed directed trail. A digraph D is trailable if D has a spanning directed trail. This dissertation focuses on a study of trailable digraphs and supereulerian digraphs from the following aspects. 1. Strong Trail-Connected, Supereulerian and Trailable Digraphs. For a digraph D, D is trailable digraph if D has a spanning trail. A digraph D is strongly trail- connected if for any two vertices u and v of D, D posses both a spanning (u, v)-trail and a spanning (v,u)-trail. As the case when u = v is possible, every strongly trail-connected digraph is also su- pereulerian. Let D be a digraph. Let S(D) = {e ∈ A(D) : e is symmetric in D}. A digraph D is symmetric if A(D) = S(D). The symmetric core of D, denoted by J(D), has vertex set V (D) and arc set S(D). We have found a well-characterized digraph family D each of whose members does not have a spanning trail with its underlying graph spanned by a K2,n−2 such that for any strong digraph D with its matching number α′(D) and arc-strong-connectivity λ(D), if n = |V (D)| ≥ 3 and λ(D) ≥ α′(D) − 1, then each of the following holds. (i) There exists a family D of well-characterized digraphs such that for any digraph D with α′(D) ≤ 2, D has a spanning trial if and only if D is not a member in D. (ii) If α′(D) ≥ 3, then D has a spanning trail. (iii) If α′(D) ≥ 3 and n ≥ 2α′(D) + 3, then D is supereulerian. (iv) If λ(D) ≥ α′(D) ≥ 4 and n ≥ 2α′(D) + 3, then for any pair of vertices u and v of D, D contains a spanning (u, v)-trail. 2. Supereulerian Digraph Strong Products. A cycle vertex cover of a digraph D is a collection of directed cycles in D such that every vertex in D lies in at least one dicycle in this collection, and such that the union of the arc sets of these directed cycles induce a connected subdigraph of D. A subdigraph F of a digraph D is a circulation if for every vertex v in F, the indegree of v equals its outdegree, and a spanning circulation if F is a cycle factor. Define f(D) to be the smallest cardinality of a cycle vertex cover of the digraph D/F obtained from D by contracting all arcs in F , among all circulations F of D. In [International Journal of Engineering Science Invention, 8 (2019) 12-19], it is proved that if D1 and D2 are nontrivial strong digraphs such that D1 is supereulerian and D2 has a cycle vertex cover C′ with |C′| ≤ |V (D1)|, then the Cartesian product D1 and D2 is also supereulerian. We prove that for strong digraphs D1 and D2, if for some cycle factor F1 of D1, the digraph formed from D1 by contracting arcs in F1 is hamiltonian with f(D2) not bigger than |V (D1)|, then the strong product D1 and D2 is supereulerian

Sink-Stable Sets of Digraphs

We introduce the notion of sink-stable sets of a digraph and prove a min-max formula for the maximum cardinality of the union of k sink-stable sets. The results imply a recent min-max theorem of Abeledo and Atkinson on the Clar number of bipartite plane graphs and a sharpening of Minty's coloring theorem. We also exhibit a link to min-max results of Bessy and Thomasse and of Sebo on cyclic stable sets

Planare Graphen und ihre Dualgraphen auf Zylinderoberflächen

In this thesis, we investigates plane drawings of undirected and directed graphs on cylinder surfaces. In the case of undirected graphs, the vertices are positioned on a line that is parallel to the cylinder’s axis and the edge curves must not intersect this line. We show that a plane drawing is possible if and only if the graph is a double-ended queue (deque) graph, i. e., the vertices of the graph can be processed according to a linear order and the edges correspond to items in the deque inserted and removed at their end vertices. A surprising consequence resulting from these observations is that the deque characterizes planar graphs with a Hamiltonian path. This result extends the known characterization of planar graphs with a Hamiltonian cycle by two stacks. By these insights, we also obtain a new characterization of queue graphs and their duals. We also consider the complexity of deciding whether a graph is a deque graph and prove that it is NP-complete. By introducing a split operation, we obtain the splittable deque and show that it characterizes planarity. For the proof, we devise an algorithm that uses the splittable deque to test whether a rotation system is planar. In the case of directed graphs, we study upward plane drawings where the edge curves follow the direction of the cylinder’s axis (standing upward planarity; SUP) or they wind around the axis (rolling upward planarity; RUP). We characterize RUP graphs by means of their duals and show that RUP and SUP swap their roles when considering a graph and its dual. There is a physical interpretation underlying this characterization: A SUP graph is to its RUP dual graph as electric current passing through a conductor to the magnetic field surrounding the conductor. Whereas testing whether a graph is RUP is NP-hard in general [Bra14], for directed graphs without sources and sink, we develop a linear-time recognition algorithm that is based on our dual graph characterization of RUP graphs.Die Arbeit beschäftigt sich mit planaren Zeichnungen ungerichteter und gerichteter Graphen auf Zylinderoberflächen. Im ungerichteten Fall werden Zeichnungen betrachtet, bei denen die Knoten auf einer Linie parallel zur Zylinderachse positioniert werden und die Kanten diese Linie nicht schneiden dürfen. Es kann gezeigt werden, dass eine planare Zeichnung genau dann möglich ist, wenn die Kanten des Graphen in einer double-ended queue (Deque) verarbeitet werden können. Ebenso lassen sich dadurch Queue, Stack und Doppelstack charakterisieren. Eine überraschende Konsequenz aus diesen Erkenntnissen ist, dass die Deque genau die planaren Graphen mit Hamiltonpfad charakterisiert. Dies erweitert die bereits bekannte Charakterisierung planarer Graphen mit Hamiltonkreis durch den Doppelstack. Im gerichteten Fall müssen die Kantenkurven entweder in Richtung der Zylinderachse verlaufen (SUP-Graphen) oder sich um die Achse herumbewegen (RUP-Graphen). Die Arbeit charakterisiert RUP-Graphen und zeigt, dass RUP und SUP ihre Rollen tauschen, wenn man Graph und Dualgraph betrachtet. Der SUP-Graph verhält sich dabei zum RUP-Graphen wie elektrischer Strom durch einen Leiter zum induzierten Magnetfeld. Ausgehend von dieser Charakterisierung ist es möglich einen Linearzeit-Algorithmus zu entwickeln, der entscheidet ob ein gerichteter Graph ohne Quellen und Senken ein RUP-Graph ist, während der allgemeine Fall NP-hart ist [Bra14]