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Real Number Graph Labeling with Distance Conditions
 SIAM J. DISCRETE MATH
, 2005
"... The theory of integer λlabellings of a graph, introduced by Griggs and Yeh, seeks to model efficient channel assignments for a network of transmitters. To prevent interference, labels for nearby vertices must be separated by specified amounts ki depending on the distance i, 1 ≤ i ≤ p. Here we expan ..."
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The theory of integer λlabellings of a graph, introduced by Griggs and Yeh, seeks to model efficient channel assignments for a network of transmitters. To prevent interference, labels for nearby vertices must be separated by specified amounts ki depending on the distance i, 1 ≤ i ≤ p. Here we expand the model to allow real number labels and separations. The main finding (“Dset Theorem”) is that for any graph, possibly infinite, with maximum degree at most ∆, there is a labelling of minimum span in which all of the labels have the form �p i=1 aiki, where the ai’s are integers ≥ 0. We show that the minimum span is a continuous function of the ki’s, and we conjecture that it is piecewise linear with finitely many pieces. Our stronger conjecture is that the coefficients ai can be bounded by a constant depending only on ∆ and p. We offer results in strong support of the conjectures, and we give formulas for the minimum spans of several graphs with general conditions at distance two. Keywords channel assignment, graph labelling, generalized coloring 1 Integer Labellings with Distance Conditions A steadily growing body of literature has evolved in the past 15 years on efficient integer labellings of the vertices of a finite simple graph with restrictions not only on adjacent vertices–as is the case with traditional graph coloring–but also on vertices at distance two.
Strengthened Brooks’ Theorem for digraphs of girth at least three
, 2011
"... Brooks’ Theorem states that a connected graph G of maximum degree ∆ has chromatic number at most ∆, unless G is an odd cycle or a complete graph. A result of Johansson shows that if G is trianglefree, then the chromatic number drops to O(∆ / log ∆). In this paper, we derive a weak analog for the ch ..."
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Brooks’ Theorem states that a connected graph G of maximum degree ∆ has chromatic number at most ∆, unless G is an odd cycle or a complete graph. A result of Johansson shows that if G is trianglefree, then the chromatic number drops to O(∆ / log ∆). In this paper, we derive a weak analog for the chromatic number of digraphs. We show that every (loopless) digraph D without directed cycles of length two has chromatic number χ(D) ≤ (1−e −13) ˜ ∆, where ˜ ∆ is the maximum geometric mean of the outdegree and indegree of a vertex in D, when ˜ ∆ is sufficiently large. As a corollary it is proved that there exists an absolute constant α < 1 such that χ(D) ≤ α ( ˜ ∆ + 1) for every ˜ ∆> 2.
Two results on the digraph chromatic number
 Discrete Mathematics
, 2012
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A Dynamic View of Circular Colorings
, 2006
"... The main contributions of this paper are threefold. First, we use a dynamic approach based on Reiter’s pioneering work on KarpMiller computation graphs [19] to give a new and short proof of Mohar’s Mintytype Theorem [15]. Second, we bridge circular colorings and discrete event dynamic systems to ..."
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The main contributions of this paper are threefold. First, we use a dynamic approach based on Reiter’s pioneering work on KarpMiller computation graphs [19] to give a new and short proof of Mohar’s Mintytype Theorem [15]. Second, we bridge circular colorings and discrete event dynamic systems to show that the Barbosa and Gafni’s results on circular chromatic number [5, 21] can be generalized to edgeweighted symmetric directed graphs. Third, we use the abovementioned dynamic view of circular colorings to construct new improved lower bounds on the circular chromatic number of a graph. We show as an example that the circular chromatic number of the line graph of the Petersen graph can be determined very easily by using these bounds. 1
EIGENVALUES AND COLORINGS OF DIGRAPHS
, 2009
"... Wilf’s eigenvalue upper bound on the chromatic number is extended to the setting of digraphs. The proof uses a generalization of Brooks’ Theorem to digraph colorings. ..."
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Wilf’s eigenvalue upper bound on the chromatic number is extended to the setting of digraphs. The proof uses a generalization of Brooks’ Theorem to digraph colorings.
Coloring Weighted SeriesParallel Graphs
, 2003
"... Let G be a seriesparallel graph with integer edge weights. A pcoloring of G is a mapping of vertices of G into Zp (ring of integers modulo p) so that the distance between colors of adjacent vertices u and v is at least the weight of the edge uv. We describe a quadratic time pcoloring algorithm wh ..."
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Let G be a seriesparallel graph with integer edge weights. A pcoloring of G is a mapping of vertices of G into Zp (ring of integers modulo p) so that the distance between colors of adjacent vertices u and v is at least the weight of the edge uv. We describe a quadratic time pcoloring algorithm where p is either twice the maximum edge weight or the largest possible sum of three weights of edges lying on a common cycle. Povzetek: Opisano je barvanje grafov. 1
unknown title
, 2003
"... Acyclic homomorphisms and circular colorings of digraphs ..."
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Weighted Coloring based Channel Assignment for WLANs
"... Abstract — We propose techniques to improve the usage of wireless spectrum in the context of wireless local area networks (WLANs) using new channel assignment methods among interfering Access Points (APs). We identify new ways of channel reuse that are based on realistic interference scenarios in W ..."
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Abstract — We propose techniques to improve the usage of wireless spectrum in the context of wireless local area networks (WLANs) using new channel assignment methods among interfering Access Points (APs). We identify new ways of channel reuse that are based on realistic interference scenarios in WLAN environments. We formulate a weighted variant of the graph coloring problem that takes into account realistic channel interference observed in wireless environments, as well as the impact of such interference on wireless users. We prove that the weighted graph coloring problem is NPhard and propose scalable distributed algorithms that achieve significantly better performance than existing techniques for channel assignment. We evaluate our algorithms through extensive simulations and experiments over an inbuilding wireless testbed. I.
Circular colorings, orientations, and weighted digraphs
, 2006
"... In this paper we prove that if a weighted symmetric digraph ( ⃗ G,c) has a mapping T: E ( ⃗ G) → {0,1} with T(xy) + T(yx) = 1 for all arcs xy in ⃗ G such that for each dicycle C satisfying 0 < Cc(mod r) < max xy∈E ( ⃗ G) c(xy) + c(yx) we have Cc/CT ≤ r, then ( ⃗ G,c) has a circular ..."
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In this paper we prove that if a weighted symmetric digraph ( ⃗ G,c) has a mapping T: E ( ⃗ G) → {0,1} with T(xy) + T(yx) = 1 for all arcs xy in ⃗ G such that for each dicycle C satisfying 0 < Cc(mod r) < max xy∈E ( ⃗ G) c(xy) + c(yx) we have Cc/CT ≤ r, then ( ⃗ G,c) has a circular rcoloring. Our result generalizes the work of Zhu (J. Comb. Theory, Ser. B, 86 (2002) 109113) concerning the (k,d)coloring of a graph, and thus is also a generalization of a corresponding result of Tuza (J. Comb. Theory, Ser. B, 55 (1992) 236243). Our result also strengthens a result of Goddyn, Tarsi and Zhang (J. Graph Theory 28 (1998) 155161) concerning the relation between orientation and the (k,d)coloring of a graph.
Coloring Digraphs with Forbidden Cycles
"... Let k and r be two integers with k 2 and k r 1. In this paper we show that (1) if a strongly connected digraph D contains no directed cycle of length 1 modulo k, then D is kcolorable; and (2) if a digraph D contains no directed cycle of length r modulo k, then D can be vertexcolored with k colo ..."
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Let k and r be two integers with k 2 and k r 1. In this paper we show that (1) if a strongly connected digraph D contains no directed cycle of length 1 modulo k, then D is kcolorable; and (2) if a digraph D contains no directed cycle of length r modulo k, then D can be vertexcolored with k colors so that each color class induces an acyclic subdigraph in D. The rst result gives an affirmative answer to a question posed by Tuza in 1992, and the second implies the following strong form of a conjecture of Diwan, Kenkre and Vishwanathan: If an undirected graph G contains no cycle of length r modulo k, then G is kcolorable if r ̸ = 2 and (k+1)colorable otherwise. Our results also strengthen several classical theorems on graph coloring proved by Bondy, Erd}os and Hajnal, Gallai and Roy, Gyarfas, etc.