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Improved bounds for some facially constrained colorings [article]

Kenny Štorgel
2020 arXiv   pre-print
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

Kenny Štorgel
2020 Discussiones Mathematicae Graph Theory  
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]

Nicolas Bousquet, Bastien Durain
2020 arXiv   pre-print
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

Sarah-Marie Belcastro, Mathematical Staircase, Inc. Holyoke MA, and Smith College, Northampton MA, USA
2018 Electronic Journal of Graph Theory and Applications  
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

Xuding Zhu
1999 Journal of combinatorial theory. Series B (Print)  
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

Ahmad Biniaz, Prosenjit Bose, Aurélien Ooms, Sander Verdonschot
2019 Graphs and Combinatorics  
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]

Ahmad Biniaz, Prosenjit Bose, Aurélien Ooms, Sander Verdonschot
2018 arXiv   pre-print
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

Borut Lužar, Jakub Przybyło, Roman Soták
2018 Journal of combinatorial optimization  
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]

Jonathan Turner
2015 arXiv   pre-print
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

C. J. Casselgren, Petros A. Petrosyan
2019 Discrete Mathematics & Theoretical Computer Science  
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]

C.J. Casselgren, Petros A. Petrosyan
2019 arXiv   pre-print
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

Benny Sudakov, Jan Volec
2017 Journal of combinatorial theory. Series B (Print)  
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]

Christopher Duffy, Fabien Jacques, Mickael Montassier, Alexandre Pinlou
2020 arXiv   pre-print
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

Geir Agnarsson, Magnús M. Halldórsson
2010 Discussiones Mathematicae Graph Theory  
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

Thamer Alsulaiman, Sushil K. Prasad, Alexander Zelikovsky
2013 International Journal of Networking and Computing  
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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