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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.  ...

### 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  ...

### 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 .  ...

### 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  ...

### 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 alight edge'' in planar graphs with minimum degree 3, which follows easily from Euler Formula.  ...

### 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.  ...

### 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.  ...

### 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.  ...

### 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.  ...

### 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.  ...

### 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.  ...

### 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.  ...

### 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.  ...