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Graph homomorphisms: structure and symmetry
"... This paper is the first part of an introduction to the subject of graph homomorphism in the mixed form of a course and a survey. We give the basic definitions, examples and uses of graph homomorphisms and mention some results that consider the structure and some parameters of the graphs involved. We ..."
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Cited by 45 (2 self)
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This paper is the first part of an introduction to the subject of graph homomorphism in the mixed form of a course and a survey. We give the basic definitions, examples and uses of graph homomorphisms and mention some results that consider the structure and some parameters of the graphs involved. We discuss vertex transitive graphs and Cayley graphs and their rather fundamental role in some aspects of graph homomorphisms. Graph colourings are then explored as homomorphisms, followed by a discussion of various graph products.
Circular colorings of edgeweighted graphs
 J. Graph Theory
, 2003
"... The notion of (circular) colorings of edgeweighted graphs is introduced. This notion generalizes the notion of (circular) colorings of graphs, the channel assignment problem, and several other optimization problems. For instance, its restriction to colorings of weighted complete graphs corresponds ..."
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Cited by 12 (4 self)
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The notion of (circular) colorings of edgeweighted graphs is introduced. This notion generalizes the notion of (circular) colorings of graphs, the channel assignment problem, and several other optimization problems. For instance, its restriction to colorings of weighted complete graphs corresponds to the traveling salesman problem (metric case). It also gives rise to a new definition of the chromatic number of directed graphs. Several basic results about the circular chromatic number of edgeweighted graphs are derived. 1
Haj'os Theorem for Colorings of EdgeWeighted Graphs
 Combinatorica
, 2001
"... Haj'os theorem states that every graph with chromatic number at least k can be obtained from the complete graph K k by a sequence of simple operations such that every intermediate graph also has chromatic number at least k. Here, Haj'os theorem is extended in three slightly different w ..."
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Cited by 5 (1 self)
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Haj'os theorem states that every graph with chromatic number at least k can be obtained from the complete graph K k by a sequence of simple operations such that every intermediate graph also has chromatic number at least k. Here, Haj'os theorem is extended in three slightly different ways to colorings and circular colorings of edgeweighted graphs. These extensions shed some new light on the Haj'os theorem and show that colorings of edgeweighted graphs are most natural extension of usual graph colorings. 1
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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Cited by 1 (1 self)
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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
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.