First László Székely and more recently Saharon Shelah and Alexander Soifer have presented examples of infinite graphs whose chromatic numbers depend on the axioms chosen for set theory. The existence of such graphs may be relevant to the Chromatic Number of the Plane problem. In this paper we construct a new class of graphs with ambiguous chromatic number. They are unit distance graphs with ver...

A faithful (unit) distance graph in Rd is a graph whose set of vertices is a finite subset of the d-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly 1. A (unit) distance graph in Rd is any subgraph of such a graph. In the first part of the paper we focus on the differences between these two classes of graphs. In particula...

We present a randomized one-round, single-bit messages, distributed algorithm for the maximum independent set problem in polynomially bounded-independence graphs with poly-logarithmic approximation factor. Bounded-independence graphs capture various models of wireless networks such as the unit disc graphs model or the quasi unit disc graphs model. For instance, on unit disc graphs, our achieved...

In 1969, Roberts introduced proper and unit interval graphs and proved that these classes are equal. Natural generalizations of unit interval graphs called k-length interval graphs were considered in which the number of different lengths of intervals is limited by k. Even after decades of research, no insight into their structure is known and the complexity of recognition is open even for k = 2...

We present a simple 3-approximation algorithm for minimum-weight dominating set and minimum-weight vertex cover in unit-disk graphs and quasi unit-disk graphs in which each node knows its coordinates. The algorithm is local: the output of a node depends solely on the input within its constantradius neighbourhood. The local horizon of the algorithm is small, both in the worst case and on average.

We show that finding a minimal number of landmark nodes for a unique virtual addressing by hop-distances in wireless ad-hoc sensor networks is NPcomplete even if the networks are unit disk graphs that contain only Gabriel edges. This problem is equivalent to Metric Dimension for Gabriel unit disk graphs. The Gabriel edges of a unit disc graph induce a planar O( √ n) distance and an optimal ener...

4 Obtaining semidefinite programs 20 4.1 Unit distance graphs and orthogonal representations . . . . . . . . . . . . . . . . . . 20 4.2 Discrete linear and quadratic programs . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.3 Spectra of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.4 Engineering applications . . . . . . . . . . . . . . . . . . ....

This article studies a degree-bounded generalization of independent sets called co-k-plexes. Constant factor approximation algorithms are developed for the maximum co-k-plex problem on unit-disk graphs. The related problem of minimum co-k-plex coloring that generalizes classical vertex coloring is also studied in the context of unit-disk graphs. We extend several classical approximation results...

An L(2, 1)-labelling of a graph G is an assignment of non-negative integers to the vertices of G such that vertices at distance at most two get different numbers and adjacent vertices get numbers which are at least two apart. The L(2, 1)-labelling number of G, denoted by λ(G), is the minimum range of labels over all such labellings. In this paper, we first discuss some necessary and sufficient ...

A unit bar-visibility graph is a graph whose vertices can be represented in the plane by disjoint horizontal unit-length bars such that two vertices are adjacent if and only if there is a unobstructed, non-degenerate, vertical band of visibility between the corresponding bars. We generalize unit bar-visibility graphs to [1, k]bar-visibility graphs by allowing the lengths of the bars to be betwe...