scholarly journals Flexible Color Lists in Alon and Tarsi's Theorem, and Time Scheduling with Unreliable Participants

10.37236/285 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Winning Strategy ◽  
Real Life ◽  
Bipartite Graphs ◽  
Upper Bounds ◽  
Combinatorial Proof ◽  
Complete Graphs ◽  
Scheduling Problem ◽  
Chromatic Numbers ◽  
List Colorings ◽  
Time Scheduling

We present a purely combinatorial proof of Alon and Tarsi's Theorem about list colorings and orientations of graphs. More precisely, we describe a winning strategy for Mrs. Correct in the corresponding coloring game of Mr. Paint and Mrs. Correct. This strategy produces correct vertex colorings, even if the colors are taken from lists that are not completely fixed before the coloration process starts. The resulting strengthening of Alon and Tarsi's Theorem leads also to strengthening of its numerous repercussions. For example we study upper bounds for list chromatic numbers of bipartite graphs and list chromatic indices of complete graphs. As real life application, we examine a chess tournament time scheduling problem with unreliable participants.

scholarly journals On star coloring of Mycielskians

2018 ◽  
Vol 2 (2) ◽  
pp. 82
Author(s):  
K. Kaliraj ◽  
V. Kowsalya ◽  
Vernold Vivin
Keyword(s):  
Chromatic Number ◽  
Graph Transformation ◽  
Clique Number ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Chromatic Numbers ◽  
Complete Bipartite Graphs ◽  
Star Coloring

<p>In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph <span class="math"><em>G</em></span> into a new graph <span class="math"><em>μ</em>(<em>G</em>)</span>, we now call the Mycielskian of <span class="math"><em>G</em></span>, which has the same clique number as <span class="math"><em>G</em></span> and whose chromatic number equals <span class="math"><em>χ</em>(<em>G</em>) + 1</span>. In this paper, we find the star chromatic number for the Mycielskian graph of complete graphs, paths, cycles and complete bipartite graphs.</p>

Roulette Wheel Selection based Heuristic Algorithm for the Orienteering Problem

2014 ◽  
Vol 13 (1) ◽  
pp. 4127-4145
Author(s):  
Madhushi Verma ◽  
Mukul Gupta ◽  
Bijeeta Pal ◽  
Prof. K. K. Shukla
Keyword(s):  
Triangle Inequality ◽  
Time Budget ◽  
Control Point ◽  
Hamiltonian Path ◽  
Real Life ◽  
Tourism Industry ◽  
Orienteering Problem ◽  
Complete Graphs ◽  
Roulette Wheel Selection ◽  
Roulette Wheel

Orienteering problem (OP) is an NP-Hard graph problem. The nodes of the graph are associated with scores or rewards and the edges with time delays. The goal is to obtain a Hamiltonian path connecting the two necessary check points, i.e. the source and the target along with a set of control points such that the total collected score is maximized within a specified time limit. OP finds application in several fields like logistics, transportation networks, tourism industry, etc. Most of the existing algorithms for OP can only be applied on complete graphs that satisfy the triangle inequality. Real-life scenario does not guarantee that there exists a direct link between all control point pairs or the triangle inequality is satisfied. To provide a more practical solution, we propose a stochastic greedy algorithm (RWS_OP) that uses the roulette wheel selectionmethod, does not require that the triangle inequality condition is satisfied and is capable of handling both complete as well as incomplete graphs. Based on several experiments on standard benchmark data we show that RWS_OP is faster, more efficient in terms of time budget utilization and achieves a better performance in terms of the total collected score ascompared to a recently reported algorithm for incomplete graphs.

scholarly journals On-Shelf Utility Mining of Sequence Data

2021 ◽  
Vol 16 (2) ◽  
pp. 1-31
Author(s):  
Chunkai Zhang ◽  
Zilin Du ◽  
Yuting Yang ◽  
Wensheng Gan ◽  
Philip S. Yu
Keyword(s):  
High Efficiency ◽  
Sequence Data ◽  
Real Life ◽  
Search Space ◽  
Upper Bounds ◽  
Utility Mining ◽  
Limited Memory ◽  
Time Periods ◽  
High Utility ◽  
Synthetic Datasets

Utility mining has emerged as an important and interesting topic owing to its wide application and considerable popularity. However, conventional utility mining methods have a bias toward items that have longer on-shelf time as they have a greater chance to generate a high utility. To eliminate the bias, the problem of on-shelf utility mining (OSUM) is introduced. In this article, we focus on the task of OSUM of sequence data, where the sequential database is divided into several partitions according to time periods and items are associated with utilities and several on-shelf time periods. To address the problem, we propose two methods, OSUM of sequence data (OSUMS) and OSUMS + , to extract on-shelf high-utility sequential patterns. For further efficiency, we also design several strategies to reduce the search space and avoid redundant calculation with two upper bounds time prefix extension utility ( TPEU ) and time reduced sequence utility ( TRSU ). In addition, two novel data structures are developed for facilitating the calculation of upper bounds and utilities. Substantial experimental results on certain real and synthetic datasets show that the two methods outperform the state-of-the-art algorithm. In conclusion, OSUMS may consume a large amount of memory and is unsuitable for cases with limited memory, while OSUMS + has wider real-life applications owing to its high efficiency.

scholarly journals On randomly 3-axial graphs

1982 ◽  
Vol 25 (2) ◽  
pp. 187-206
Author(s):  
Yousef Alavi ◽  
Sabra S. Anderson ◽  
Gary Chartrand ◽  
S.F. Kapoor
Keyword(s):  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Complete Graphs ◽  
Initial Vertex ◽  
Complete Bipartite Graphs

A graph G, every vertex of which has degree at least three, is randomly 3-axial if for each vertex v of G, any ordered collection of three paths in G of length one with initial vertex v can be cyclically randomly extended to produce three internally disjoint paths which contain all the vertices of G. Randomly 3-axial graphs of order p > 4 are characterized for p ≢ 1 (mod 3), and are shown to be either complete graphs or certain regular complete bipartite graphs.

scholarly journals Solving the nuclear dismantling project scheduling problem by combining mixed-integer and constraint programming techniques and metaheuristics

2021 ◽  
Author(s):  
Felix Hübner ◽  
Patrick Gerhards ◽  
Christian Stürck ◽  
Rebekka Volk
Keyword(s):  
Constraint Programming ◽  
Project Scheduling ◽  
Computational Study ◽  
Real Life ◽  
Mixed Integer ◽  
Extensive Literature ◽  
Scheduling Problem ◽  
Multiple Modes ◽  
Project Scheduling Problem ◽  
Optimisation Techniques

AbstractScheduling of megaprojects is very challenging because of typical characteristics, such as expected long project durations, many activities with multiple modes, scarce resources, and investment decisions. Furthermore, each megaproject has additional specific characteristics to be considered. Since the number of nuclear dismantling projects is expected to increase considerably worldwide in the coming decades, we use this type of megaproject as an application case in this paper. Therefore, we consider the specific characteristics of constrained renewable and non-renewable resources, multiple modes, precedence relations with and without no-wait condition, and a cost minimisation objective. To reliably plan at minimum costs considering all relevant characteristics, scheduling methods can be applied. But the extensive literature review conducted did not reveal a scheduling method considering the special characteristics of nuclear dismantling projects. Consequently, we introduce a novel scheduling problem referred to as the nuclear dismantling project scheduling problem. Furthermore, we developed and implemented an effective metaheuristic to obtain feasible schedules for projects with about 300 activities. We tested our approach with real-life data of three different nuclear dismantling projects in Germany. On average, it took less than a second to find an initial feasible solution for our samples. This solution could be further improved using metaheuristic procedures and exact optimisation techniques such as mixed-integer programming and constraint programming. The computational study shows that utilising exact optimisation techniques is beneficial compared to standard metaheuristics. The main result is the development of an initial solution finding procedure and an adaptive large neighbourhood search with iterative destroy and recreate operations that is competitive with state-of-the-art methods of related problems. The described problem and findings can be transferred to other megaprojects.

ON MOSTAR INDEX OF GRAPHS

2021 ◽  
Vol 10 (4) ◽  
pp. 2115-2129
Author(s):  
P. Kandan ◽  
S. Subramanian
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Bipartite Graphs ◽  
Topological Indices ◽  
Great Success ◽  
Complete Graphs ◽  
Molecular Graphs ◽  
Graphs And Networks ◽  
Complete Bipartite Graphs ◽  
Simple Connected Graph

On the great success of bond-additive topological indices like Szeged, Padmakar-Ivan, Zagreb, and irregularity measures, yet another index, the Mostar index, has been introduced recently as a peripherality measure in molecular graphs and networks. For a connected graph G, the Mostar index is defined as $$M_{o}(G)=\displaystyle{\sum\limits_{e=gh\epsilon E(G)}}C(gh),$$ where $C(gh) \,=\,\left|n_{g}(e)-n_{h}(e)\right|$ be the contribution of edge $uv$ and $n_{g}(e)$ denotes the number of vertices of $G$ lying closer to vertex $g$ than to vertex $h$ ($n_{h}(e)$ define similarly). In this paper, we prove a general form of the results obtained by $Do\check{s}li\acute{c}$ et al.\cite{18} for compute the Mostar index to the Cartesian product of two simple connected graph. Using this result, we have derived the Cartesian product of paths, cycles, complete bipartite graphs, complete graphs and to some molecular graphs.

scholarly journals Metaheuristics for Order Scheduling Problem with Unequal Ready Times

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Jan-Yee Kung ◽  
Jiahui Duan ◽  
Jianyou Xu ◽  
I-Hong Chung ◽  
Shuenn-Ren Cheng ◽  
...  
Keyword(s):  
Real Life ◽  
Experimental Tests ◽  
Service Systems ◽  
Scheduling Problem ◽  
Due Dates ◽  
Order Scheduling ◽  
Customer Order ◽  
Scheduling Model ◽  
Unequal Ready Times ◽  
Customer Order Scheduling

In recent years, various customer order scheduling (OS) models can be found in numerous manufacturing and service systems in which several designers, who have developed modules independently for several different products, convene as a product development team, and that team completes a product design only after all the modules have been designed. In real-life situations, a customer order can have some requirements including due dates, weights of jobs, and unequal ready times. Once encountering different ready times, waiting for future order or job arrivals to raise the completeness of a batch is an efficient policy. Meanwhile, the literature releases that few studies have taken unequal ready times into consideration for order scheduling problem. Motivated by this limitation, this study addresses an OS scheduling model with unequal order ready times. The objective function is to find a schedule to optimize the total completion time criterion. To solve this problem for exact solutions, two lower bounds and some properties are first derived to raise the searching power of a branch-and-bound method. For approximate solution, four simulated annealing approaches and four heuristic genetic algorithms are then proposed. At last, several experimental tests and their corresponding statistical outcomes are also reported to examine the performance of all the proposed methods.

scholarly journals Can Colour-Blind Distinguish Colour Palettes?

10.37236/2319 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Rafał Kalinowski ◽  
Monika Pilśniak ◽  
Jakub Przybyło ◽  
Mariusz Woźniak
Keyword(s):  
Large Degree ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Regular Graphs ◽  
Lovász Local Lemma ◽  
Local Lemma ◽  
Comprehensive Information ◽  
Lovasz Local Lemma ◽  
Delta G ◽  
Irregular Graphs

Let $c:E(G)\rightarrow [k]$ be  a colouring, not necessarily proper, of edges of a graph $G$. For a vertex $v\in V$, let $\overline{c}(v)=(a_1,\ldots,a_k)$, where $ a_i =|\{u:uv\in E(G),\;c(uv)=i\}|$, for $i\in [k].$ If we re-order the sequence $\overline{c}(v)$ non-decreasingly, we obtain a sequence $c^*(v)=(d_1,\ldots,d_k)$, called a palette of a vertex $v$. This can be viewed as the most comprehensive information about colours incident with $v$ which can be delivered by a person who is unable to name colours but distinguishes one from another. The smallest $k$ such that $c^*$ is a proper colouring of vertices of $G$ is called the colour-blind index of a graph $G$, and is denoted by dal$(G)$. We conjecture that there is a constant $K$ such that dal$(G)\leq K$ for every graph $G$ for which the parameter is well defined. As our main result we prove that $K\leq 6$ for regular graphs of sufficiently large degree, and for irregular graphs with $\delta (G)$ and $\Delta(G)$ satisfying certain conditions. The proofs are based on the Lopsided Lovász Local Lemma. We also show that $K=3$ for all regular bipartite graphs, and for complete graphs of order $n\geq 8$.

scholarly journals Consecutive Colouring of Oriented Graphs

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Marta Borowiecka-Olszewska ◽  
Ewa Drgas-Burchardt ◽  
Nahid Yelene Javier-Nol ◽  
Rita Zuazua
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Oriented Graphs ◽  
Complete Problem ◽  
Np Complete

AbstractWe consider arc colourings of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals. We prove that the existence of such a colouring is an NP-complete problem. We give the solution of the problem for r-regular oriented graphs, transitive tournaments, oriented graphs with small maximum degree, oriented graphs with small order and some other classes of oriented graphs. We state the conjecture that for each graph there exists a consecutive colourable orientation and confirm the conjecture for complete graphs, 2-degenerate graphs, planar graphs with girth at least 8, and bipartite graphs with arboricity at most two that include all planar bipartite graphs. Additionally, we prove that the conjecture is true for all perfect consecutively colourable graphs and for all forbidden graphs for the class of perfect consecutively colourable graphs.

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