183 research outputs found
Graph Coloring Problems and Group Connectivity
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
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
The \emph{separation dimension} of a graph , written , is the minimum number of linear orderings of 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} , which is the minimum of such that some linear orderings (repetition allowed) separate every two nonincident edges at least times.
In contrast to separation dimension, we show fractional separation dimension is bounded: always , with equality if and only if contains . There is no stronger bound even for bipartite graphs, since . We also compute for cycles and some complete tripartite graphs. We show that when is a tree and present a sequence of trees on which the value tends to . We conjecture that when the -free -vertex graph maximizing is .
We also consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let be the number of circular orderings needed to separate all pairs, and let be the fractional version. Among our results: (1) if and only is outerplanar. (2) when is bipartite. (3) . (4) , with equality if and only if . (5) .
A \emph{star -coloring} is a proper -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 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 is \emph{-dynamic} if for each , at least colors appear in . We investigate -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 -coloring of a graph , we define the \emph{sum value} of a vertex to be . The smallest integer such that has a proper total -coloring whose sum values form a proper coloring is the \emph{neighbor sum distinguishing total chromatic number} . Pil{\'s}niak and Wo{\'z}niak~(2013) conjectured that for any simple graph with maximum degree . We prove this bound to be asymptotically correct by showing that . 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
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 -stable locally checkable problems parameterized by mim-width
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 -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 -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 -stable locally checkable problems
whose complexity on graphs of bounded mim-width was open so far
Twin-width VIII: delineation and win-wins
We introduce the notion of delineation. A graph class is said
delineated if for every hereditary closure of a subclass of
, it holds that has bounded twin-width if and only if
is monadically dependent. An effective strengthening of
delineation for a class implies that tractable FO model checking
on is perfectly understood: On hereditary closures of
subclasses of , FO model checking is fixed-parameter tractable
(FPT) exactly when 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 -free segment graphs, and
axis-parallel -free unit segment graphs have bounded twin-width, where
is the half-graph or ladder of height . In contrast, axis-parallel
-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 -sequences, lead to win-win arguments. For instance, we derive
FPT algorithms for -Ladder on visibility graphs of 1.5D terrains, and
-Independent Set on visibility graphs of simple polygons.Comment: 51 pages, 19 figure
Symmetry in Graph Theory
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
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