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Improved bounds for some facially constrained colorings
[article]
2020
arXiv
pre-print
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A facial-parity edge-coloring of a 2-edge-connected plane graph is a facially-proper edge-coloring in which every face is incident with zero or an odd number of edges of each color. ...
Czap and Jendroľ (in Facially-constrained colorings of plane graphs: A survey, Discrete Math. 340 (2017), 2691–2703), conjectured that 10 colors suffice in both colorings. ...
In regards with facial WORM vertex-coloring of plane graphs, it is known that not all plane graphs have a (P 3 , P 3 )-WORM coloring. ...
arXiv:2005.09979v1
fatcat:uz4lpmywjnffvldtjgj5gwspli
Improved bounds for some facially constrained colorings
2020
Discussiones Mathematicae Graph Theory
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A facial-parity edge-coloring of a 2-edge-connected plane graph is a facially-proper edge-coloring in which every face is incident with zero or an odd number of edges of each color. ...
Czap and Jendroľ in [Facially-constrained colorings of plane graphs: A survey, Discrete Math. 340 (2017) 2691-2703], conjectured that 10 colors suffice in both colorings. ...
Improved Bounds for Some Facially Constrained Colorings ...
doi:10.7151/dmgt.2357
fatcat:lkmfcv473bfw5ltje6kv5ddct4
A note on the simultaneous edge coloring
[article]
2020
arXiv
pre-print
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Recently, Cabello raised the following question: given two graphs G_1,G_2 of maximum degree Δ on the same set of vertices V, is it possible to edge-color their (edge) union with Δ+2 colors in such a way ...
,G_ℓ of maximum degree Δ with Ω(√(ℓ)·Δ) colors and that there exist graphs for which this bound is tight up to a constant multiplicative factor. ...
Since edges of H 2 are in both G 1 and G 2 , the vertex v has degree at most ∆ − d in both graphs H 1 1 and H 2 1 . ...
arXiv:2001.01463v1
fatcat:vhz4nnj4cnbydoy7y7kc2rcduu
Parsimonious edge-coloring on surfaces
2018
Electronic Journal of Graph Theory and Applications
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We correct a small error in a 1996 paper of Albertson and Haas, and extend their lower bound for the fraction of properly colorable edges of planar subcubic graphs that are simple, connected, bridgeless ...
, and edge-maximal to other surface embeddings of subcubic graphs. ...
Acknowledgement The author is grateful to Mike Albertson for suggesting that she examine this problem, and to both Mike Albertson and Ruth Haas for discussions (both long-ago and recent) of their paper ...
doi:10.5614/ejgta.2018.6.2.9
fatcat:55bnzvtnsngxxmgrvtowjrteze
The Game Coloring Number of Planar Graphs
1999
Journal of combinatorial theory. Series B (Print)
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This paper discusses a variation of the game chromatic number of a graph: the game coloring number. This parameter provides an upper bound for the game chromatic number of a graph. ...
We show that the game coloring number of a planar graph is at most 19. ...
The proof in this paper only uses a very simple property of planar graphs: the existence of a``light edge'' in planar graphs with minimum degree 3, which follows easily from Euler Formula. ...
doi:10.1006/jctb.1998.1878
fatcat:opm6xhs3qrgmthlldvdtqapcme
Improved Bounds for Guarding Plane Graphs with Edges
2019
Graphs and Combinatorics
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bound of 3n 8 edges for any plane graph. 2. ...
An edge guard set of a plane graph G is a subset Γ of edges of G such that each face of G is incident to an endpoint of an edge in Γ. Such a set is said to guard G. ...
We need to color its vertices with two colors, say white and blue, such that every face contains (i) vertices of both colors and (ii) an edge whose endpoints have the same color. ...
doi:10.1007/s00373-018-02004-z
fatcat:aoutdfmy3ffz7jztaji2vhhliq
Improved Bounds for Guarding Plane Graphs with Edges
[article]
2018
arXiv
pre-print
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that G can be guarded with at most 2n/5 edges, then extend this approach with a deeper analysis to yield an improved bound of 3n/8 edges for any plane graph. 2- We prove that there exists an edge guard ...
set of G with at most n/3+α/9 edges, where α is the number of quadrilateral faces in G. ...
We need to color its vertices with two colors, say white and blue, such that every face contains (i) vertices of both colors and (ii) an edge whose endpoints have the same color. ...
arXiv:1804.07150v1
fatcat:yrda5xoe2zd7dpadcgw3ohqaku
New bounds for locally irregular chromatic index of bipartite and subcubic graphs
2018
Journal of combinatorial optimization
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2016 pre-print arXiv:1611.02341v1 fatcat:3rlwqktq6rabholshjyhfpmig4Other Versions
A graph is locally irregular if the neighbors of every vertex v have degrees distinct from the degree of v. locally irregular edge-coloring of a graph G is an (improper) edge-coloring such that the graph ...
In addition, we also prove that 4 colors suffice for locally irregular edge-coloring of any subcubic graph. ...
Lužar also acknowledges partial support by the National Scholarship Programme of the Slovak Republic. ...
doi:10.1007/s10878-018-0313-7
fatcat:vh3zc5npzbe3dogktfsdapbmo4
The Bounded Edge Coloring Problem and Offline Crossbar Scheduling
[article]
2015
arXiv
pre-print
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This paper introduces a variant of the classical edge coloring problem in graphs that can be applied to an offline scheduling problem for crossbar switches. ...
We show that the problem is NP-complete, develop three lower bounds bounds on the optimal solution value and evaluate the performance of several approximation algorithms, both analytically and experimentally ...
For each edge in E that is colored k − 1, we include a chain of three edges, with the inner edge assigned a lower bound of k, while the outer two edges are assigned lower bounds of k − 1. ...
arXiv:1512.09002v1
fatcat:omkxexamqfhbhgk7yxw7pmaxlq
Some results on the palette index of graphs
2019
Discrete Mathematics & Theoretical Computer Science
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Given a proper edge coloring $\varphi$ of a graph $G$, we define the palette $S_{G}(v,\varphi)$ of a vertex $v \in V(G)$ as the set of all colors appearing on edges incident with $v$. ...
In this paper we give various upper and lower bounds on the palette index of $G$ in terms of the vertex degrees of $G$, particularly for the case when $G$ is a bipartite graph with small vertex degrees ...
Acknowledgements The authors would like to thank the referees for helpful comments and suggestions, particularly for pointing out an argument which simplified the proof of Theorem 3.6. ...
doi:10.23638/dmtcs-21-3-11
fatcat:cizc65lcore5tc3lczny75cjdm
Some results on the palette index of graphs
[article]
2019
arXiv
pre-print
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Given a proper edge coloring φ of a graph G, we define the palette S_G(v,φ) of a vertex v ∈ V(G) as the set of all colors appearing on edges incident with v. ...
In this paper we give various upper and lower bounds on the palette index of G in terms of the vertex degrees of G, particularly for the case when G is a bipartite graph with small vertex degrees. ...
Acknowledgements The authors would like to thank the referees for helpful comments and suggestions, particularly for pointing out an argument which simplified the proof of Theorem 3.6. ...
arXiv:1805.00260v4
fatcat:prrti4o63vc5joqhahqrkttdgm
Properly colored and rainbow copies of graphs with few cherries
2017
Journal of combinatorial theory. Series B (Print)
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We also prove that one can find a rainbow copy of such G in every edge-coloring of K_n in which all colors appear bounded number of times. ...
Let G be an n-vertex graph that contains linearly many cherries (i.e., paths on 3 vertices), and let c be a coloring of the edges of the complete graph K_n such that at each vertex every color appears ...
If the two other vertices v 1 and v 2 in P z are both neighbors of z in T 3n , then the vertices of P z span a monochromatic cherry and the copy is not properly edge-colored. ...
doi:10.1016/j.jctb.2016.07.001
fatcat:2f5makrhizazbir67atnk5wbsa
The chromatic number of 2-edge-colored and signed graphs of bounded maximum degree
[article]
2020
arXiv
pre-print
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A 2-edge-colored graph or a signed graph is a simple graph with two types of edges. ...
A homomorphism from a 2-edge-colored graph G to a 2-edge-colored graph H is a mapping φ: V(G) → V(H) that maps every edge in G to an edge of the same type in H. ...
in both kinds of graphs: with a positive or a negative edge in the case of 2-edge-colored graphs, with a toward or a backward arc in the oriented case. ...
arXiv:2009.05439v1
fatcat:v7orrs2u3re25momwmj4qrrbqi
Vertex coloring the square of outerplanar graphs of low degree
2010
Discussiones Mathematicae Graph Theory
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In this article we prove that the chromatic number of the square of an outerplanar graph of maximum degree ∆ = 6 is 7. ...
The optimal upper bound for the chromatic number of the square of an outerplanar graph of maximum degree ∆ = 6 is known. ...
Acknowledgments The authors are grateful to Steve Hedetniemi for his interest in this problem and for suggesting the writing of this article. ...
doi:10.7151/dmgt.1518
fatcat:nu2vdbfpmzh2pnogbbdiphgkdi
Distributed Algorithms for TDMA Link Scheduling in Sensor Networks
2013
International Journal of Networking and Computing
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In this paper, our contribution includes the following: Formulating FDLSP as distance-2 edge coloring in bi-directed graphs. ...
We formulate the full duplex link scheduling problem (FDLSP) as distance-2 edge coloring in bi-directed graphs and prove tighter lower and upper bounds for the FDLSP problem. ...
An Upper Bound Lemma 6. Any bi-directed graph G with the maximum out-degree ∆ requires at most 2∆ 2 colors to distance-2 edge-color graph G. Proof. ...
doi:10.15803/ijnc.3.1_55
fatcat:tk4gbj6xtrbyfftizcxcfmqzfq
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