Synchronizing Graphs

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Graph Theory ◽  
Graph Coloring ◽  
Edge Coloring ◽  
Chromatic Index ◽  
Ramsey Numbers ◽  
Edge Colorings ◽  
The Road ◽  
New Type ◽  
Traffic Systems ◽  
Coloring Theorem

This chapter considers a new type of graph coloring known as edge coloring. It begins with a discussion of an idea by Scottish physicist Peter Guthrie Tait that led to edge coloring. Tait proved that the regions of every 3-regular bridgeless planar graph could be colored with four or fewer colors if and only if the edges of such a graph could be colored with three colors so that every two adjacent edges are colored differently. Tait thought that he had found a new way to solve the Four Color Problem. The chapter also examines the chromatic index of a graph, Vizing's Theorem, applications of edge colorings, and a class of numbers in graph theory called Ramsey numbers. Finally, it describes the Road Coloring Theorem which deals with traffic systems consisting only of one-way streets in which the same number of roads leave each location.

Generalized edge-colorings of weighted graphs

2016 ◽  
Vol 08 (01) ◽  
pp. 1650015
Author(s):  
Yuji Obata ◽  
Takao Nishizeki
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Edge Coloring ◽  
Efficient Algorithms ◽  
Upper And Lower Bounds ◽  
Weighted Graphs ◽  
Chromatic Index ◽  
Edge Colorings ◽  
Integer Weight

Let [Formula: see text] be a graph with a positive integer weight [Formula: see text] for each vertex [Formula: see text]. One wishes to assign each edge [Formula: see text] of [Formula: see text] a positive integer [Formula: see text] as a color so that [Formula: see text] for any vertex [Formula: see text] and any two edges [Formula: see text] and [Formula: see text] incident to [Formula: see text]. Such an assignment [Formula: see text] is called an [Formula: see text]-edge-coloring of [Formula: see text], and the maximum integer assigned to edges is called the span of [Formula: see text]. The [Formula: see text]-chromatic index of [Formula: see text] is the minimum span over all [Formula: see text]-edge-colorings of [Formula: see text]. In the paper, we present various upper and lower bounds on the [Formula: see text]-chromatic index, and obtain three efficient algorithms to find an [Formula: see text]-edge-coloring of a given graph. One of them finds an [Formula: see text]-edge-coloring with span smaller than twice the [Formula: see text]-chromatic index.

scholarly journals A note on compact and compact circular edge-colorings of graphs

2008 ◽  
Vol Vol. 10 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Dariusz Dereniowski ◽  
Adam Nadolski
Keyword(s):  
Approximation Algorithm ◽  
Decision Problem ◽  
Edge Coloring ◽  
Exact Algorithm ◽  
Weighted Graphs ◽  
Edge Colorings ◽  
Odd Cycle ◽  
International Audience ◽  
Np Complete ◽  
Odd Cycles

Graphs and Algorithms International audience We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.

Adjacent vertex distinguishing edge coloring of planar graphs without 3-cycles

2020 ◽  
Vol 12 (04) ◽  
pp. 2050035
Author(s):  
Danjun Huang ◽  
Xiaoxiu Zhang ◽  
Weifan Wang ◽  
Stephen Finbow
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Edge Coloring ◽  
Discrete Math ◽  
Adjacent Vertex ◽  
Chromatic Index ◽  
Proper Edge Coloring ◽  
Minimum Number

The adjacent vertex distinguishing edge coloring of a graph [Formula: see text] is a proper edge coloring of [Formula: see text] such that the color sets of any pair of adjacent vertices are distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of [Formula: see text] is denoted by [Formula: see text]. It is observed that [Formula: see text] when [Formula: see text] contains two adjacent vertices of degree [Formula: see text]. In this paper, we prove that if [Formula: see text] is a planar graph without 3-cycles, then [Formula: see text]. Furthermore, we characterize the adjacent vertex distinguishing chromatic index for planar graphs of [Formula: see text] and without 3-cycles. This improves a result from [D. Huang, Z. Miao and W. Wang, Adjacent vertex distinguishing indices of planar graphs without 3-cycles, Discrete Math. 338 (2015) 139–148] that established [Formula: see text] for planar graphs without 3-cycles.

A CHANNEL ASSIGNMENT PROBLEM IN MULTIHOP WIRELESS NETWORKS AND GRAPH THEORY

2004 ◽  
Vol 13 (02) ◽  
pp. 375-385 ◽  
Author(s):  
HIROSHI TAMURA ◽  
KAORU WATANABE ◽  
MASAKAZU SENGOKU ◽  
SHOJI SHINODA
Keyword(s):  
Wireless Networks ◽  
Graph Theory ◽  
Channel Assignment ◽  
Assignment Problem ◽  
Edge Coloring ◽  
Multihop Wireless Networks ◽  
Personal Communication ◽  
Multihop Wireless ◽  
Coloring Problem ◽  
Communication Devices

Multihop wireless networks consist of mobile terminals with personal communication devices. Each terminal can receive a message and then send it to another terminal. In these networks, it is important to assign channels for communications to each terminal efficiently. There are some studies on this assignment problem using a conventional edge coloring in graph theory. In this paper, we propose a new edge coloring problem in graph and network theory on this assignment problem and we discuss the computational complexity of the problem. This edge coloring problem takes the degree of interference into consideration. Therefore, we can reuse the channels more efficiently compared with the conventional method.

scholarly journals A survey of graph coloring - its types, methods and applications

2012 ◽  
Vol 37 (3) ◽  
pp. 223-238 ◽  
Author(s):  
Piotr Formanowicz ◽  
Krzysztof Tanaś
Keyword(s):  
Graph Theory ◽  
Graph Coloring ◽  
Cubic Graphs ◽  
The World ◽  
Computer Scientists

Abstract Graph coloring is one of the best known, popular and extensively researched subject in the field of graph theory, having many applications and conjectures, which are still open and studied by various mathematicians and computer scientists along the world. In this paper we present a survey of graph coloring as an important subfield of graph theory, describing various methods of the coloring, and a list of problems and conjectures associated with them. Lastly, we turn our attention to cubic graphs, a class of graphs, which has been found to be very interesting to study and color. A brief review of graph coloring methods (in Polish) was given by Kubale in [32] and a more detailed one in a book by the same author. We extend this review and explore the field of graph coloring further, describing various results obtained by other authors and show some interesting applications of this field of graph theory.

scholarly journals Rainbow Connection Number and Chromatic Index of Rough Ideal based Rough Edge Cayley Graph

2018 ◽  
Vol 7 (3) ◽  
pp. 1926 ◽  
Author(s):  
B. Praba ◽  
X.A. Benazir Obilia
Keyword(s):  
Graph Theory ◽  
Cayley Graph ◽  
Chromatic Index ◽  
Connection Number ◽  
Rainbow Connection Number ◽  
Rainbow Connection

Rainbow connection number and chromatic index are two significant parameters in the study ofgraph theory. In this work, rainbow connection number and chromatic index of Rough Ideal based Rough Edge Cayley Graph G(T(J)) are evaluated. We prove that the rainbow connection number of G(T(J)) is 2 and the chromatic index of G(T(J)) is 2(2n^m)(3m^1):Rainbow connection number and chromatic index are two significant parameters in the study of graph theory. In this work, rainbow connection number and chromatic index of Rough Ideal based Rough Edge Cayley Graph  are evaluated. We prove that the rainbow connection number of  is 2 and the chromatic index of  is .

Analysis of Preference for Autonomous Driving Under Different Traffic Conditions Using a Driving Simulator

2015 ◽  
Vol 27 (6) ◽  
pp. 660-670 ◽  
Author(s):  
Udara Eshan Manawadu ◽  
◽  
Masaaki Ishikawa ◽  
Mitsuhiro Kamezaki ◽  
Shigeki Sugano ◽  
...  
Keyword(s):  
Autonomous Vehicles ◽  
Driving Simulator ◽  
Autonomous Driving ◽  
Driving Experience ◽  
The Road ◽  
Traffic Conditions ◽  
Passenger Vehicles ◽  
New Type ◽  
On The Road ◽  
Near Future

<div class=""abs_img""><img src=""[disp_template_path]/JRM/abst-image/00270006/08.jpg"" width=""300"" /> Driving simulator</div>Intelligent passenger vehicles with autonomous capabilities will be commonplace on our roads in the near future. These vehicles will reshape the existing relationship between the driver and vehicle. Therefore, to create a new type of rewarding relationship, it is important to analyze when drivers prefer autonomous vehicles to manually-driven (conventional) vehicles. This paper documents a driving simulator-based study conducted to identify the preferences and individual driving experiences of novice and experienced drivers of autonomous and conventional vehicles under different traffic and road conditions. We first developed a simplified driving simulator that could connect to different driver-vehicle interfaces (DVI). We then created virtual environments consisting of scenarios and events that drivers encounter in real-world driving, and we implemented fully autonomous driving. We then conducted experiments to clarify how the autonomous driving experience differed for the two groups. The results showed that experienced drivers opt for conventional driving overall, mainly due to the flexibility and driving pleasure it offers, while novices tend to prefer autonomous driving due to its inherent ease and safety. A further analysis indicated that drivers preferred to use both autonomous and conventional driving methods interchangeably, depending on the road and traffic conditions.

On the Adjacent Vertex Distinguishing Proper Edge Colorings of Several Classes of Complete 5-Partite Graphs

2013 ◽  
Vol 333-335 ◽  
pp. 1452-1455
Author(s):  
Chun Yan Ma ◽  
Xiang En Chen ◽  
Fang Yang ◽  
Bing Yao
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Adjacent Vertex ◽  
Chromatic Numbers ◽  
Edge Colorings ◽  
Proper Edge Coloring ◽  
Minimum Number

A proper $k$-edge coloring of a graph $G$ is an assignment of $k$ colors, $1,2,\cdots,k$, to edges of $G$. For a proper edge coloring $f$ of $G$ and any vertex $x$ of $G$, we use $S(x)$ denote the set of thecolors assigned to the edges incident to $x$. If for any two adjacent vertices $u$ and $v$ of $G$, we have $S(u)\neq S(v)$,then $f$ is called the adjacent vertex distinguishing proper edge coloring of $G$ (or AVDPEC of $G$ in brief). The minimum number of colors required in an AVDPEC of $G$ is called the adjacent vertex distinguishing proper edge chromatic number of $G$, denoted by $\chi^{'}_{\mathrm{a}}(G)$. In this paper, adjacent vertex distinguishing proper edge chromatic numbers of several classes of complete 5-partite graphs are obtained.

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