On the independence ratio of distance graphs

A distance graph is an undirected graph on the integers where two integers are adjacent if their difference is in a prescribed distance set. The independence ratio of a distance graph $G$ is the maximum density of an independent set in $G$. Lih, Liu, and Zhu [Star extremal circulant graphs, SIAM J. Discrete Math. 12 (1999) 491--499] showed that the independence ratio is equal to the inverse of the fractional chromatic number, thus relating the concept to the well studied question of finding the chromatic number of distance graphs. We prove that the independence ratio of a distance graph is achieved by a periodic set, and we present a framework for discharging arguments to demonstrate upper bounds on the independence ratio. With these tools, we determine the exact independence ratio for several infinite families of distance sets of size three, determine asymptotic values for others, and present several conjectures.Comment: 39 pages, 12 figures, 6 table

On the Circular Chromatic Number of Circular Partitionable Graphs

International audienceThis paper studies the circular chromatic number of a class of circular partitionable graphs. We prove that an infinite family of circular partitionable graphs Ghas Xc (G) = X (G). A consequence of this result is that we obtain an infinite family of graphs G with the rare property that the deletion of each vertex decreases its circular chromatic number by exactly

4-Colorable 6-regular toroidal graphs

AbstractThis paper proves two conjectures of Collins, Fisher and Hutchinson about the chromatic number of some circulant graphs. As a consequence, we characterize 4-colorable 6-regular toroidal graphs

Defensive alliances in regular graphs and circulant graphs

In this paper we study defensive alliances in some regular graphs. We determine which subgraphs could a critical defensive alliance of a graph $G$ induce, if $G$ is $6$-regular and the cardinality of the alliance is at most $8$. In particular, we study the case of circulant graphs, i.e. Cayley graphs on a cyclic group. The critical defensive alliances of a circulant graph of degree at most $6$ are completely determined. For the general case, we give tight lower and upper bounds on the alliance number of a circulant graph with $d$ generators.Preprin

On Motzkin’s problem in the circle group

The version of record is available online at: https://doi.org/10.1134/S0081543821040039Given a subset D of the interval (0,1), if a Borel set A¿[0,1) contains no pair of elements whose difference modulo 1 is in D, then how large can the Lebesgue measure of A be? This is the analogue in the circle group of a well-known problem of Motzkin, originally posed for sets of integers. We make a first treatment of this circle-group analogue, for finite sets D of missing differences, using techniques from ergodic theory, graph theory and the geometry of numbers. Our results include an exact solution when D has two elements at least one of which is irrational. When every element of D is rational, the problem is equivalent to estimating the independence ratio of a circulant graph. In the case of two rational elements, we give an estimate for this ratio in terms of the odd girth of the graph, which is asymptotically sharp and also recovers the classical solution of Cantor and Gordon to Motzkin’s original problem for two missing differences.Peer ReviewedPostprint (published version

Scheduling links for heavy traffic on interfering routes in wireless mesh networks

We consider wireless mesh networks and the problem of scheduling the links of a given set of routes under the assumption of a heavy-traffic pattern. We assume some TDMA protocol provides a background of synchronized time slots and seek to schedule the routes' links to maximize the number of packets that get delivered to their destinations per time slot. Our approach is to construct an undirected graph G and to heuristically obtain node multicolorings for G that can be turned into efficient link schedules. In G each node represents a link to be scheduled and the edges are set up to represent every possible interference for any given set of interference assumptions. We present two multicoloring-based heuristics and study their performance through extensive simulations. One of the two heuristics is based on relaxing the notion of a node multicoloring by dynamically exploiting the availability of communication opportunities that would otherwise be wasted. We have found that, as a consequence, its performance is significantly superior to the other's

Edge and total colourings of graphs

Die vorliegenden Arbeit enthält Ergebnisse zu Kanten- und Totalfärbungen von Graphen sowie verschiedenen Variationen dieser Färbungen. Eine Kantenfärbung eines Graphen G ist eine Zuordnung von Farben zu den Kanten von G, so dass adjazente Kanten unterschiedliche Farben erhalten. Eine Totalfärbung ist eine Färbung der Knoten und Kanten von G, so dass adjazente Knoten, adjazente Kanten sowie ein Knoten und eine inzidente Kante jeweils unterschiedlich gefärbt werden. Der chromatische Index bzw. die totalchromatische Zahl von G bezeichnen die kleinste Anzahl von Farben, mit denen G kantenfärbbar bzw. totalfärbbar ist. In dieser Arbeit wird unter anderem die totalchromatische Zahl zirkulanter Graphen mit Maximalgrad 3 bestimmt sowie ein Algorithmus entwickelt, der alle planaren kritischen Graphen der Kantenfärbung mit bis zu 12 Knoten konstruiert und darstellt. Das Konzept der Kreisfärbung von Graphen wird von Knoten- auf Kanten- und Totalfärbung übertragen; Eigenschaften des kreischromatischen Index und der kreistotalchromatischen Zahl werden bewiesen und exakte Werte für einige Graphenklassen ermittelt. Die listenchromatische Vermutung wird für outerplanare Graphen mit Maximalgrad >4 bewiesen. Die Konzepte der (a,b)- und (a,b,r)-Listen- färbung werden von Knotenfärbung auf Kantenfärbung übertragen; es werden Eigenschaften dieser Färbungen und Ergebnisse für einzelne Graphenklassen hergeleitet.This thesis contains results for edge and total colourings as well as for some variations of these colourings. An edge colouring of a graph G is an assignment of colours to the edges of G such that adjacent edges are coloured differently. A total colouring is a colouring of the vertices and edges of G such that adjacent vertices, adjacent edges as well as a vertex and an incident edge are coloured differently. The chromatic index or the total chromatic number of G denote the minimum number of colours such that G admits an edge colouring or a total colouring, respectively. Results in this thesis are - among others - the total chromatic number of circulant graphs with maximum degree 3 and an algorithm to construct and draw all planar critical graphs with at most 12 vertices. The concept of circular colourings is transferred from vertex to edge and total colourings. Properties of the circular chromatic index and the circular total chromatic number are proven and exact values are determined for some classes of graphs. The list chromatic conjecture is confirmed for outerplanar graphs with maximum degree >4; the concepts of (a,b)- and (a,b,r)-list colourings are transferred from vertex to edge colouring and properties of these colourings as well as results for special classes of graphs are given