scholarly journals Rainbow Connectivity Using a Rank Genetic Algorithm: Moore Cages with Girth Six

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
J. Cervantes-Ojeda ◽  
M. Gómez-Fuentes ◽  
D. González-Moreno ◽  
M. Olsen
Keyword(s):  
Genetic Algorithm ◽  
Upper Bound ◽  
Connected Graph ◽  
Edge Coloring ◽  
Np Hard ◽  
The Family ◽  
Minimum Number ◽  
The One ◽  
Vertex Disjoint ◽  
Rainbow Coloring

Arainbowt-coloringof at-connected graphGis an edge coloring such that for any two distinct verticesuandvofGthere are at leasttinternally vertex-disjoint rainbow(u,v)-paths. In this work, we apply a Rank Genetic Algorithm to search for rainbowt-colorings of the family of Moore cages with girth six(t;6)-cages. We found that an upper bound in the number of colors needed to produce a rainbow 4-coloring of a(4;6)-cage is 7, improving the one currently known, which is 13. The computation of the minimum number of colors of a rainbow coloring is known to be NP-Hard and the Rank Genetic Algorithm showed good behavior finding rainbowt-colorings with a small number of colors.

scholarly journals Graph coloring using the reduced quantum genetic algorithm

2022 ◽  
Vol 7 ◽  
pp. e836
Author(s):  
Sebastian Mihai Ardelean ◽  
Mihai Udrescu
Keyword(s):  
Genetic Algorithm ◽  
Graph Coloring ◽  
Optimization Problems ◽  
Edge Coloring ◽  
Np Hard ◽  
Quantum Genetic Algorithm ◽  
Graph Coloring Problem ◽  
Minimum Number ◽  
Hard Problems ◽  
Noise Models

Genetic algorithms (GA) are computational methods for solving optimization problems inspired by natural selection. Because we can simulate the quantum circuits that implement GA in different highly configurable noise models and even run GA on actual quantum computers, we can analyze this class of heuristic methods in the quantum context for NP-hard problems. This paper proposes an instantiation of the Reduced Quantum Genetic Algorithm (RQGA) that solves the NP-hard graph coloring problem in O(N1/2). The proposed implementation solves both vertex and edge coloring and can also determine the chromatic number (i.e., the minimum number of colors required to color the graph). We examine the results, analyze the algorithm convergence, and measure the algorithm's performance using the Qiskit simulation environment. Our Reduced Quantum Genetic Algorithm (RQGA) circuit implementation and the graph coloring results show that quantum heuristics can tackle complex computational problems more efficiently than their conventional counterparts.

A NOTE ON GENERALIZED RAINBOW CONNECTION OF CONNECTED GRAPHS AND THEIR NUMBER OF EDGES

2021 ◽  
Vol 66 (3) ◽  
pp. 3-7
Author(s):  
Anh Nguyen Thi Thuy ◽  
Duyen Le Thi
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Connection Number ◽  
Connected Graphs ◽  
Rainbow Connection Number ◽  
Rainbow Connection ◽  
Rainbow Path ◽  
Vertex Disjoint

Let l ≥ 1, k ≥ 1 be two integers. Given an edge-coloured connected graph G. A path P in the graph G is called l-rainbow path if each subpath of length at most l + 1 is rainbow. The graph G is called (k, l)-rainbow connected if any two vertices in G are connected by at least k pairwise internally vertex-disjoint l-rainbow paths. The smallest number of colours needed in order to make G (k, l)-rainbow connected is called the (k, l)-rainbow connection number of G and denoted by rck,l(G). In this paper, we first focus to improve the upper bound of the (1, l)-rainbow connection number depending on the size of connected graphs. Using this result, we characterize all connected graphs having the large (1, 2)-rainbow connection number. Moreover, we also determine the (1, l)-rainbow connection number in a connected graph G containing a sequence of cut-edges.

scholarly journals ON SPANNERS OF GEOMETRIC GRAPHS

2009 ◽  
Vol 20 (01) ◽  
pp. 135-149 ◽  
Author(s):  
JOACHIM GUDMUNDSSON ◽  
MICHIEL SMID
Keyword(s):  
Real Number ◽  
Upper Bound ◽  
Weighted Graph ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Np Hard ◽  
Minimum Number ◽  
Np Hardness

Given a connected geometric graph G, we consider the problem of constructing a t-spanner of G having the minimum number of edges. We prove that for every real number t with [Formula: see text], there exists a connected geometric graph G with n vertices, such that every t-spanner of G contains Ω(n1+1/t) edges. This bound almost matches the known upper bound, which states that every connected weighted graph with n vertices contains a t-spanner with O(n1+2/(t-1)) edges. We also prove that the problem of deciding whether a given geometric graph contains a t-spanner with at most K edges is NP-hard. Previously, this NP-hardness result was only known for non-geometric graphs.

Tight Upper Bound of the 3-Total-Rainbow Index for 2-(Edge-)Connected Graphs

2021 ◽  
pp. 2142001
Author(s):  
Yingbin Ma ◽  
Wenhan Zhu
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Total Coloring ◽  
Colored Graph ◽  
Fixed Integer ◽  
Connected Graphs ◽  
Vertex Set ◽  
Rainbow Coloring

Let [Formula: see text] be an edge-colored graph with order [Formula: see text] and [Formula: see text] be a fixed integer satisfying [Formula: see text]. For a vertex set [Formula: see text] of at least two vertices, a tree containing the vertices of [Formula: see text] in [Formula: see text] is called an [Formula: see text]-tree. The [Formula: see text]-tree [Formula: see text] is a total-rainbow [Formula: see text]-tree if the elements of [Formula: see text], except for the vertex set [Formula: see text], have distinct colors. A total-colored graph [Formula: see text] is said to be total-rainbow [Formula: see text]-tree connected if for every set [Formula: see text] of [Formula: see text] vertices in [Formula: see text], there exists a total-rainbow [Formula: see text]-tree in [Formula: see text], while the total-coloring of [Formula: see text] is called a [Formula: see text]-total-rainbow coloring. The [Formula: see text]-total-rainbow index of a nontrivial connected graph [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed in a [Formula: see text]-total-rainbow coloring of [Formula: see text]. In this paper, we show a sharp upper bound for [Formula: see text], where [Formula: see text] is a 2-connected or 2-edge-connected graph.

scholarly journals An Explicit Edge-Coloring of $K_n$ with Six Colors on Every $K_5$

10.37236/7852 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Alex Cameron
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Edge Coloring ◽  
Minimum Number ◽  
Positive Integers

Let $p$ and $q$ be positive integers such that $1 \leq q \leq {p \choose 2}$. A $(p,q)$-coloring of the complete graph on $n$ vertices $K_n$ is an edge coloring for which every $p$-clique contains edges of at least $q$ distinct colors. We denote the minimum number of colors needed for such a $(p,q)$-coloring of $K_n$ by $f(n,p,q)$. This is known as the Erdös-Gyárfás function. In this paper we give an explicit $(5,6)$-coloring with $n^{1/2+o(1)}$ colors. This improves the best known upper bound of $f(n,5,6)=O\left(n^{3/5}\right)$ given by Erdös and Gyárfás, and comes close to matching the order of the best known lower bound, $f(n,5,6) = \Omega\left(n^{1/2}\right)$.

scholarly journals Edge coloring of graphs, uses, limitation, complexity

2016 ◽  
Vol 8 (1) ◽  
pp. 63-81
Author(s):  
Sándor Szabó ◽  
Bogdán Zaválnij
Keyword(s):  
Upper Bound ◽  
Search Algorithm ◽  
Edge Coloring ◽  
Clique Number ◽  
Optimal Number ◽  
Np Hard ◽  
Main Motivation

Abstract The known fact that coloring of the nodes of a graph improves the performance of practical clique search algorithm is the main motivation of this paper. We will describe a number of ways in which an edge coloring scheme proposed in [8] can be used in clique search. The edge coloring provides an upper bound for the clique number. This estimate has a limitation. It will be shown that the gap between the clique number and the upper bound can be arbitrarily large. Finally, it will be shown that to determine the optimal number of colors in an edge coloring is NP-hard.

scholarly journals TWO RESULTS ON THE PALETTE INDEX OF GRAPHS

2021 ◽  
Vol 55 (1 (254)) ◽  
pp. 36-43
Author(s):  
Khachik S. Smbatyan
Keyword(s):  
Bipartite Graph ◽  
Upper Bound ◽  
Edge Coloring ◽  
Cartesian Products ◽  
Proper Edge Coloring ◽  
Minimum Number ◽  
Nearly Bipartite Graph ◽  
Cartesian Products Of Graphs

Given a proper edge coloring $\alpha$ of a graph $G$, we define the palette $S_G(v,\alpha)$ of a vertex $v\in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $\check{s}(G)$ of $G$ is the minimum number of distinct palettes occurring in a proper edge coloring of $G$. A graph $G$ is called nearly bipartite if there exists $ v\in V(G)$ so that $G-v$ is a bipartite graph. In this paper, we give an upper bound on the palette index of a nearly bipartite graph $G$ by using the decomposition of $G$ into cycles. We also provide an upper bound on the palette index of Cartesian products of graphs. In particular, we show that for any graphs $G$ and $H$, $\check{s}(G\square H)\leq \check{s}(G)\check{s}(H)$.

scholarly journals The Chromatic Index of a Graph Whose Core has Maximum Degree $2$

10.37236/2101 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Mikio Kano ◽  
Saieed Akbari ◽  
Maryam Ghanbari ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Edge Coloring ◽  
Chromatic Index ◽  
The Core ◽  
Class 1 ◽  
Minimum Number ◽  
Even Order ◽  
Delta G

Let $G$ be a graph. The core of $G$, denoted by $G_{\Delta}$, is the subgraph of $G$ induced by the vertices of degree $\Delta(G)$, where $\Delta(G)$ denotes the maximum degree of $G$. A $k$-edge coloring of $G$ is a function $f:E(G)\rightarrow L$ such that $|L| = k$ and $f(e_1)\neq f(e_2)$ for all two adjacent edges  $e_1$ and $e_2$ of $G$. The chromatic index of $G$, denoted by $\chi'(G)$, is the minimum number $k$ for which $G$ has a $k$-edge coloring.  A graph $G$ is said to be Class $1$ if $\chi'(G) = \Delta(G)$ and Class $2$ if $\chi'(G) = \Delta(G) + 1$. In this paper it is shown that every connected graph $G$ of even order and with $\Delta(G_{\Delta})\leq 2$ is Class $1$ if $|G_{\Delta}|\leq 9$ or $G_{\Delta}$ is a cycle of order $10$.

A Remark on Rainbow 6-Cycles in Hypercubes

2018 ◽  
Vol 28 (02) ◽  
pp. 1850007
Author(s):  
Chen Hao ◽  
Weihua Yang
Keyword(s):  
Positive Integer ◽  
Edge Coloring ◽  
Minimum Number ◽  
Dimensional Cube ◽  
Cube Formula ◽  
Rainbow Coloring

We call an edge-coloring of a graph [Formula: see text] a rainbow coloring if the edges of [Formula: see text] are colored with distinct colors. For every even positive integer [Formula: see text], let [Formula: see text] denote the minimum number of colors required to color the edges of the [Formula: see text]-dimensional cube [Formula: see text], so that every copy of [Formula: see text] is rainbow. Faudree et al. [6] proved that [Formula: see text] for [Formula: see text] or [Formula: see text]. Mubayi et al. [8] showed that [Formula: see text]. In this note, we show that [Formula: see text]. Moreover, we obtain the number of 6-cycles of [Formula: see text].

ON PALETTE INDEX OF UNICYCLE AND BICYCLE GRAPHS

2019 ◽  
Vol 53 (1 (248)) ◽  
pp. 3-12
Author(s):  
A.B. Ghazaryan
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Edge Coloring ◽  
Cyclomatic Number ◽  
Proper Edge Coloring ◽  
Minimum Number ◽  
Delta G

Given a proper edge coloring $ \phi $ of a graph $ G $, we define the palette $ S_G (\nu, \phi) $ of a vertex $ \nu \mathclose{\in} V(G) $ as the set of all colors appearing on edges incident with $ \nu $. The palette index $ \check{s} (G) $ of $ G $ is the minimum number of distinct palettes occurring in a proper edge coloring of $ G $. In this paper we give an upper bound on the palette index of a graph G in terms of cyclomatic number $ cyc(G) $ of $ G $ and maximum degree $ \Delta (G) $ of $ G $. We also give a sharp upper bound for the palette index of unicycle and bicycle graphs.

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