183 research outputs found

    Graph Coloring Problems and Group Connectivity

    Get PDF
    1. Group connectivity. Let A be an abelian group and let iA(G) be the smallest positive integer m such that Lm(G) is A-connected. A path P of G is a normal divalent path if all internal vertices of P are of degree 2 in G and if |E(P)|= 2, then P is not in a 3-cycle of G. Let l(G) = max{lcub}m : G has a normal divalent path of length m{rcub}. We obtain the following result. (i) If |A| ≥ 4, then iA( G) ≤ l(G). (ii) If | A| ≥ 4, then iA(G) ≤ |V(G)| -- Delta(G). (iii) Suppose that |A| ≥ 4 and d = diam( G). If d ≤ |A| -- 1, then iA(G) ≤ d; and if d ≥ |A|, then iA(G) ≤ 2d -- |A| + 1. (iv) iZ 3 (G) ≤ l(G) + 2. All those bounds are best possible.;2. Modulo orientation. A mod (2p + 1)-orientation D is an orientation of G such that d +D(v) = d--D(v) (mod 2p + 1) for any vertex v ∈ V ( G). We prove that for any integer t ≥ 2, there exists a finite family F = F(p, t) of graphs that do not have a mod (2p + 1)-orientation, such that every graph G with independence number at most t either admits a mod (2p+1)-orientation or is contractible to a member in F. In particular, the graph family F(p, 2) is determined, and our results imply that every 8-edge-connected graph G with independence number at most two admits a mod 5-orientation.;3. Neighbor sum distinguishing total coloring. A proper total k-coloring &phis; of a graph G is a mapping from V(G) ∪ E(G) to {lcub}1,2, . . .,k{rcub} such that no adjacent or incident elements in V(G) ∪ E( G) receive the same color. Let m&phis;( v) denote the sum of the colors on the edges incident with the vertex v and the color on v. A proper total k-coloring of G is called neighbor sum distinguishing if m &phis;(u) ≠ m&phis;( v) for each edge uv ∈ E( G ). Let chitSigma(G) be the neighbor sum distinguishing total chromatic number of a graph G. Pilsniak and Wozniak conjectured that for any graph G, chitSigma( G) ≤ Delta(G) + 3. We show that if G is a graph with treewidth ℓ ≥ 3 and Delta(G) ≥ 2ℓ + 3, then chitSigma( G) + ℓ -- 1. This upper bound confirms the conjecture for graphs with treewidth 3 and 4. Furthermore, when ℓ = 3 and Delta ≥ 9, we show that Delta(G)+1 ≤ chit Sigma(G) ≤ Delta(G)+2 and characterize graphs with equalities.;4. Star edge coloring. A star edge coloring of a graph is a proper edge coloring such that every connected 2-colored subgraph is a path with at most 3 edges. Let ch\u27st(G) be the list star chromatic index of G: the minimum s such that for every s-list assignment L for the edges, G has a star edge coloring from L. By introducing a stronger coloring, we show with a very concise proof that the upper bound of the star chromatic index of trees also holds for list star chromatic index of trees, i.e. ch\u27st( T) ≤ [3Delta/2] for any tree T with maximum degree Delta. And then by applying some orientation technique we present two upper bounds for list star chromatic index of k-degenerate graphs

    Labeling of graphs, sumset of squares of units modulo n and resonance varieties of matroids

    Get PDF
    This thesis investigates problems in a number of different areas of graph theory and its applications in other areas of mathematics. Motivated by the 1-2-3-Conjecture, we consider the closed distinguishing number of a graph G, denoted by dis[G]. We provide new upper bounds for dis[G] by using the Combinatorial Nullstellensatz. We prove that it is NP-complete to decide for a given planar subcubic graph G, whether dis[G] = 2. We show that for each integer t there is a bipartite graph G such that dis[G] \u3e t. Then some polynomial time algorithms and NP-hardness results for the problem of partitioning the edges of a graph into regular and/or locally irregular subgraphs are presented. We then move on to consider Johnson graphs to find resonance varieties of some classes of sparse paving matroids. The last application we consider is in number theory, where we find the number of solutions of the equation x21 + _ _ _ + x2 k = c, where c 2 Zn, and xi are all units in the ring Zn. Our approach is combinatorial using spectral graph theory

    Coloring and covering problems on graphs

    Get PDF
    The \emph{separation dimension} of a graph GG, written π(G)\pi(G), is the minimum number of linear orderings of V(G)V(G) such that every two nonincident edges are ``separated'' in some ordering, meaning that both endpoints of one edge appear before both endpoints of the other. We introduce the \emph{fractional separation dimension} πf(G)\pi_f(G), which is the minimum of a/ba/b such that some aa linear orderings (repetition allowed) separate every two nonincident edges at least bb times. In contrast to separation dimension, we show fractional separation dimension is bounded: always πf(G)3\pi_f(G)\le 3, with equality if and only if GG contains K4K_4. There is no stronger bound even for bipartite graphs, since πf(Km,m)=πf(Km+1,m)=3mm+1\pi_f(K_{m,m})=\pi_f(K_{m+1,m})=\frac{3m}{m+1}. We also compute πf(G)\pi_f(G) for cycles and some complete tripartite graphs. We show that πf(G)<2\pi_f(G)<\sqrt{2} when GG is a tree and present a sequence of trees on which the value tends to 4/34/3. We conjecture that when n=3mn=3m the K4K_4-free nn-vertex graph maximizing πf(G)\pi_f(G) is Km,m,mK_{m,m,m}. We also consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let π(G)\pi^\circ(G) be the number of circular orderings needed to separate all pairs, and let πf(G)\pi_f^\circ(G) be the fractional version. Among our results: (1) π(G)=1\pi^\circ(G)=1 if and only GG is outerplanar. (2) π(G)2\pi^\circ(G)\le2 when GG is bipartite. (3) π(Kn)log2log3(n1)\pi^\circ(K_n)\ge\log_2\log_3(n-1). (4) πf(G)32\pi_f^\circ(G)\le\frac{3}{2}, with equality if and only if K4GK_4\subseteq G. (5) πf(Km,m)=3m32m1\pi_f^\circ(K_{m,m})=\frac{3m-3}{2m-1}. A \emph{star kk-coloring} is a proper kk-coloring where the union of any two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an \emph{\Ifp}. We use discharging to prove that every graph with maximum average degree less than 52\frac{5}{2} has an \Ifp, which is sharp and improves the result of Bu, Cranston, Montassier, Raspaud, and Wang (2009). As a corollary, we gain that every planar graph with girth at least 10 has a star 4-coloring. A proper vertex coloring of a graph GG is \emph{rr-dynamic} if for each vV(G)v\in V(G), at least min{r,d(v)}\min\{r,d(v)\} colors appear in NG(v)N_G(v). We investigate 33-dynamic versions of coloring and list coloring. We prove that planar and toroidal graphs are 3-dynamically 10-choosable, and this bound is sharp for toroidal graphs. Given a proper total kk-coloring cc of a graph GG, we define the \emph{sum value} of a vertex vv to be c(v)+uvE(G)c(uv)c(v) + \sum_{uv \in E(G)} c(uv). The smallest integer kk such that GG has a proper total kk-coloring whose sum values form a proper coloring is the \emph{neighbor sum distinguishing total chromatic number} χΣ(G)\chi''_{\Sigma}(G). Pil{\'s}niak and Wo{\'z}niak~(2013) conjectured that χΣ(G)Δ(G)+3\chi''_{\Sigma}(G)\leq \Delta(G)+3 for any simple graph with maximum degree Δ(G)\Delta(G). We prove this bound to be asymptotically correct by showing that χΣ(G)Δ(G)(1+o(1))\chi''_{\Sigma}(G)\leq \Delta(G)(1+o(1)). The main idea of our argument relies on Przyby{\l}o's proof (2014) for neighbor sum distinguishing edge-coloring

    Rooted structures in graphs: a project on Hadwiger's conjecture, rooted minors, and Tutte cycles

    Get PDF
    Hadwigers Vermutung ist eine der anspruchsvollsten Vermutungen für Graphentheoretiker und bietet eine weitreichende Verallgemeinerung des Vierfarbensatzes. Ausgehend von dieser offenen Frage der strukturellen Graphentheorie werden gewurzelte Strukturen in Graphen diskutiert. Eine Transversale einer Partition ist definiert als eine Menge, welche genau ein Element aus jeder Menge der Partition enthält und sonst nichts. Für einen Graphen G und eine Teilmenge T seiner Knotenmenge ist ein gewurzelter Minor von G ein Minor, der T als Transversale seiner Taschen enthält. Sei T eine Transversale einer Färbung eines Graphen, sodass es ein System von kanten-disjunkten Wegen zwischen allen Knoten aus T gibt; dann stellt sich die Frage, ob es möglich ist, die Existenz eines vollständigen, in T gewurzelten Minors zu gewährleisten. Diese Frage ist eng mit Hadwigers Vermutung verwoben: Eine positive Antwort würde Hadwigers Vermutung für eindeutig färbbare Graphen bestätigen. In dieser Arbeit wird ebendiese Fragestellung untersucht sowie weitere Konzepte vorgestellt, welche bekannte Ideen der strukturellen Graphentheorie um eine Verwurzelung erweitern. Beispielsweise wird diskutiert, inwiefern hoch zusammenhängende Teilmengen der Knotenmenge einen hoch zusammenhängenden, gewurzelten Minor erzwingen. Zudem werden verschiedene Ideen von Hamiltonizität in planaren und nicht-planaren Graphen behandelt.Hadwiger's Conjecture is one of the most tantalising conjectures for graph theorists and offers a far-reaching generalisation of the Four-Colour-Theorem. Based on this major issue in structural graph theory, this thesis explores rooted structures in graphs. A transversal of a partition is a set which contains exactly one element from each member of the partition and nothing else. Given a graph G and a subset T of its vertex set, a rooted minor of G is a minor such that T is a transversal of its branch set. Assume that a graph has a transversal T of one of its colourings such that there is a system of edge-disjoint paths between all vertices from T; it comes natural to ask whether such graphs contain a minor rooted at T. This question of containment is strongly related to Hadwiger's Conjecture; indeed, a positive answer would prove Hadwiger's Conjecture for uniquely colourable graphs. This thesis studies the aforementioned question and besides, presents several other concepts of attaching rooted relatedness to ideas in structural graph theory. For instance, whether a highly connected subset of the vertex set forces a highly connected rooted minor. Moreover, several ideas of Hamiltonicity in planar and non-planar graphs are discussed

    On dd-stable locally checkable problems parameterized by mim-width

    Full text link
    In this paper we continue the study of locally checkable problems under the framework introduced by Bonomo-Braberman and Gonzalez in 2020, by focusing on graphs of bounded mim-width. We study which restrictions on a locally checkable problem are necessary in order to be able to solve it efficiently on graphs of bounded mim-width. To this end, we introduce the concept of dd-stability of a check function. The related locally checkable problems contain large classes of problems, among which we can mention, for example, LCVP problems. We give an algorithm showing that these problems are XP when parameterized by the mim-width of a given binary decomposition tree of the input graph, that is, that they can be solved in polynomial time given a binary decomposition tree of bounded mim-width. We explore the relation between dd-stable locally checkable problems and the recently introduced DN logic (Bergougnoux, Dreier and Jaffke, 2022), and show that both frameworks model the same family of problems. We include a list of concrete examples of dd-stable locally checkable problems whose complexity on graphs of bounded mim-width was open so far

    Twin-width VIII: delineation and win-wins

    Get PDF
    We introduce the notion of delineation. A graph class C\mathcal C is said delineated if for every hereditary closure D\mathcal D of a subclass of C\mathcal C, it holds that D\mathcal D has bounded twin-width if and only if D\mathcal D is monadically dependent. An effective strengthening of delineation for a class C\mathcal C implies that tractable FO model checking on C\mathcal C is perfectly understood: On hereditary closures D\mathcal D of subclasses of C\mathcal C, FO model checking is fixed-parameter tractable (FPT) exactly when D\mathcal D has bounded twin-width. Ordered graphs [BGOdMSTT, STOC '22] and permutation graphs [BKTW, JACM '22] are effectively delineated, while subcubic graphs are not. On the one hand, we prove that interval graphs, and even, rooted directed path graphs are delineated. On the other hand, we show that segment graphs, directed path graphs, and visibility graphs of simple polygons are not delineated. In an effort to draw the delineation frontier between interval graphs (that are delineated) and axis-parallel two-lengthed segment graphs (that are not), we investigate the twin-width of restricted segment intersection classes. It was known that (triangle-free) pure axis-parallel unit segment graphs have unbounded twin-width [BGKTW, SODA '21]. We show that Kt,tK_{t,t}-free segment graphs, and axis-parallel HtH_t-free unit segment graphs have bounded twin-width, where HtH_t is the half-graph or ladder of height tt. In contrast, axis-parallel H4H_4-free two-lengthed segment graphs have unbounded twin-width. Our new results, combined with the known FPT algorithm for FO model checking on graphs given with O(1)O(1)-sequences, lead to win-win arguments. For instance, we derive FPT algorithms for kk-Ladder on visibility graphs of 1.5D terrains, and kk-Independent Set on visibility graphs of simple polygons.Comment: 51 pages, 19 figure

    Symmetry in Graph Theory

    Get PDF
    This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view
    corecore