Compact cyclic edge-colorings of graphs

Nadolski, Adam

doi:10.1016/j.disc.2006.09.058

Cited by 12 publications

(21 citation statements)

References 12 publications

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“…Cyclic compact k-edge-colorings are also studied in [9] and are closely related to the circular compact colorability defined in [10]. Given a positive real number r, let C r denote the circle with circumference r. Taking an arbitrary point 0 ∈ C r and …”

Section: Introductionmentioning

confidence: 99%

On compact k-edge-colorings: A polynomial time reduction from linear to cyclic

Altinakar

Caporossi

Hertz

2011

Discrete Optimization

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a b s t r a c tA k-edge-coloring of a graph G = (V , E) is a function c that assigns an integer c(e) (called color) in {0, 1, . . . , k − 1} to every edge e ∈ E so that adjacent edges get different colors.A k-edge-coloring is linear compact if the colors on the edges incident to every vertex are consecutive. The problem k-LCCP is to determine whether a given graph admits a linear compact k-edge coloring. A k-edge-coloring is cyclic compact if for every vertex v there are two positive integers a v , b v in {0, 1, . . . , k − 1} such that the colors on the edges incident to v are exactly {a v , (a v + 1)mod k, . . . , b v }. The problem k-CCCP is to determine whether a given graph admits a cyclic compact k-edge coloring. We show that the k-LCCP with possibly imposed or forbidden colors on some edges is polynomially reducible to the k-CCCP when k ≥ 12, and to the 12-CCCP when k < 12.

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Section: Introductionmentioning

confidence: 99%

On compact k-edge-colorings: A polynomial time reduction from linear to cyclic

Altinakar

Caporossi

Hertz

2011

Discrete Optimization

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“…Kubale and Nadolski showed that the problem of determining whether a given bipartite graph admits a cyclic interval coloring is

N P

‐complete. Some sufficient conditions for a graph to have a cyclic interval coloring were obtained in . Nadolski proved that any connected graph G with

Δ (G) \leq 3

has a cyclic interval coloring.…”

Section: Introductionmentioning

confidence: 99%

“…Some sufficient conditions for a graph to have a cyclic interval coloring were obtained in . Nadolski proved that any connected graph G with

Δ (G) \leq 3

has a cyclic interval coloring. de Werra and Solot proved that any outerplanar bipartite graph G has a cyclic interval

Δ (G)

‐coloring.…”

Section: Introductionmentioning

confidence: 99%

Some results on cyclic interval edge colorings of graphs

Asratian

Casselgren

2017

Journal of Graph Theory

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A proper edge coloring of a graph G with colors 1, 2, . . . , t is called a cyclic interval t-coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. We prove that a bipartite graph G of even maximum degree ∆(G) ≥ 4 admits a cyclic intervalWe also prove that every Eulerian bipartite graph G with maximum degree at most 8 has a cyclic interval coloring. Some results are obtained for (a, b)-biregular graphs, that is, bipartite graphs with the vertices in one part all having degree a and the vertices in the other part all having degree b; it has been conjectured that all these have cyclic interval colorings. We show that all (4, 7)-biregular graphs as well as all (2r −2, 2r)-biregular (r ≥ 2) graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this proves a conjecture of Petrosyan and Mkhitaryan.

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“…Cyclic interval colorings are studied in, e.g. . In particular, the general question of determining whether a bipartite graph G has a cyclic interval coloring is

scriptNP

‐complete and there are concrete examples of connected bipartite graphs having no cyclic interval coloring .…”

Section: Introductionmentioning

confidence: 99%

“…. In particular, the general question of determining whether a bipartite graph G has a cyclic interval coloring is

scriptNP

‐complete and there are concrete examples of connected bipartite graphs having no cyclic interval coloring . Trivially, any bipartite graph with an interval coloring also has a cyclic interval coloring with

Δ (G)

colors, but the converse does not hold .…”

Section: Introductionmentioning

confidence: 99%

On Interval Edge Colorings of Biregular Bipartite Graphs With Small Vertex Degrees

Casselgren

Toft

2014

Journal of Graph Theory

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Abstract. A proper edge coloring of a graph with colors 1, 2, 3, . . . is called an interval coloring if the colors on the edges incident to each vertex form an interval of integers. A bipartite graph is (a, b)-biregular if every vertex in one part has degree a and every vertex in the other part has degree b. It has been conjectured that all such graphs have interval colorings. We prove that all (3, 6)-biregular graphs have interval colorings and that all (3, 9)-biregular graphs having a cubic subgraph covering all vertices of degree 9 admit interval colorings. Moreover, we prove that slightly weaker versions of the conjecture hold for (3, 5)-biregular, (4, 6)-biregular and (4, 8)-biregular graphs. All our proofs are constructive and yield polynomial time algorithms for constructing the required colorings.

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Compact cyclic edge-colorings of graphs

Cited by 12 publications

References 12 publications

On compact k-edge-colorings: A polynomial time reduction from linear to cyclic

On compact k-edge-colorings: A polynomial time reduction from linear to cyclic

Some results on cyclic interval edge colorings of graphs

On Interval Edge Colorings of Biregular Bipartite Graphs With Small Vertex Degrees

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