Colorable Graphs Research Articles

On 3-Coloring Circle Graphs

Given a graph $G$ with a fixed vertex order $\prec$, one obtains a circle graph $H$ whose vertices are the edges of $G$ and where two such edges are adjacent if and only if their endpoints are pairwise distinct and alternate in $\prec$. Therefore, the problem of determining whether $G$ has a $k$-page book embedding with spine order $\prec$ is equivalent to deciding whether $H$ can be colored with $k$ colors. Finding a $k$-coloring for a circle graph is known to be NP-complete for $k \geq 4$ [9] and trivial for $k \leq 2$. For $k = 3$, Unger (1992) claims an efficient algorithm that finds a 3-coloring in $O(n \log n)$ time, if it exists. Given a circle graph $H$, Unger's algorithm (1) constructs a 3-\textsc{Sat} formula $\Phi$ that is satisfiable if and only if $H$ admits a 3-coloring and (2) solves $\Phi$ by a backtracking strategy that relies on the structure imposed by the circle graph. However, the extended abstract misses several details and Unger refers to his PhD thesis (in German) for details. In this paper we argue that Unger's algorithm for 3-coloring circle graphs is not correct and that 3-coloring circle graphs should be considered as an open problem. We show that step (1) of Unger's algorithm is incorrect by exhibiting a circle graph and its representation whose formula $\Phi$ is satisfiable but that is not 3-colorable. We further show that Unger's backtracking strategy for solving $\Phi$ in step (2) may produce incorrect results and give empirical evidence that it exhibits a runtime behaviour that is not consistent with the claimed running time.

Application of a novel visual representation method for urban noise source monitoring

Increasing urban mobility has led to a rise in the number of vehicles, resulting in urban noise pollution, which can cause health problems. Conventional methods for noise control can be adopted to reduce noise near the sources. However, this is not cost-effective to reduce overall noise level in the urban area. Therefore, it is preferable to encourage drivers to voluntarily reduce noise by applying methods such as cracking down on road noise sources. To this end, a real-time urban noise source monitoring system is needed, but processing large amounts of data can be difficult. In this study, a novel visual representation method, DoAgram, is applied for urban noise source monitoring. DoAgram can represent the time, frequency, and spatial information of the sound source within a 2D color graph. Therefore, the long-term spatial information of the sound source acquired from the sensor array node can be transmitted as a single image. Also, one can restore spatial information of the source from an image through a decoding process corresponding to the encoding method. One can find that the proposed method can be utilized as a database that can efficiently conduct urban noise source monitoring.

A note on interval colourings of graphs

A graph is said to be interval colourable if it admits a proper edge-colouring using palette N in which the set of colours of edges that are incident to each vertex is an interval. The interval colouring thickness of a graph G is the minimum k such that G can be edge-decomposed into k interval colourable graphs. We show that θ(n), the maximum interval colouring thickness of an n-vertex graph, satisfies θ(n)=Ω(log(n)/loglog(n)) and θ(n)⩽n5/6+o(1), which improves on the trivial lower bound and the upper bound given by the first author and Zheng. As a corollary, we answer a question of Asratian, Casselgren, and Petrosyan and disprove a conjecture of Borowiecka-Olszewska, Drgas-Burchardt, Javier-Nol, and Zuazua. We also confirm a conjecture of the first author that any interval colouring of an n-vertex planar graph uses at most 3n/2−2 colours.

Exploring luminescence in transparent Glass–Ceramic coating: Study on Pr3+/ Dy3+ Co-doped SrO–Al2O3–SiO2 Glass–Ceramic particles

This study aimed to fabricate transparent glass–ceramic coatings with luminescent characteristics. To this end, a SrO–Al2O3–SiO2 glass–ceramic system doped with Pr3+ and Dy3+ ions, referred to as luminescent glass–ceramic (LGC), was synthesized via melting at 1550 °C for 1 h. LGCs were added as mill additives to the transparent glass–ceramic coating recipe at increasing weight percentages of 5, 10, 20, and 40% (5Pr, 10Pr, 20Pr, and 40Pr, respectively). The coating specimens fabricated through the wet spraying technique were vitrified and recrystallized at 820 °C for 4 min, resulting in an LGC coating. Further, the chemical composition, phase structure, thermal behavior, and bonding characteristics of the LGC systems were analyzed, and spectrofluorometer and scanning electron microscope (SEM) analyses were performed after the coating procedure. The SEM indicated an increase in particle intensity per unit surface area. The spectrofluorometric analysis revealed that the standard sample did not display any discernible peaks; however, samples supplemented with particles exhibit emission bands at wavelengths of 485 (4F9/2 → 6H15/2, blue) and 575 nm (4F9/2 → 6H13/2, yellow) for Dy3+ ions and at 485 nm (3P0→3H4, blue) for Pr3+ ions. According to the CIE color graph, an increase in the quantity of added particles resulted in the region shifting toward the green–white area from the center of the blue–green region in the standard sample.

Decomposing planar graphs without triangular short cycles into a matching and a 3-colorable graph

Decomposing planar graphs without triangular short cycles into a matching and a 3-colorable graph

Near optimal colourability on hereditary graph families

In this paper, we initiate a systematic study on a new notion called near optimal colourability which is closely related to perfect graphs and the Lovász theta function. A graph family G is near optimal colourable if there is a constant number c such that every graph G∈G satisfies χ(G)≤max⁡{c,ω(G)}, where χ(G) and ω(G) are the chromatic number and clique number of G, respectively. The near optimal colourable graph families together with the Lovász theta function are useful for the study of the colouring problems for hereditary graph families. We investigate the near optimal colourability for (H1,H2)-free graphs. Our main result is an almost complete characterisation for the near optimal colourability for (H1,H2)-free graphs with two exceptional cases, one of which is the celebrated Gyárfás conjecture. As an application of our results, we show that the colouring problem for (2K2,P4∨Kn)-free graphs is polynomial time solvable, which solves an open problem in Dabrowski and Paulusma (2018) [6].

Burning, edge burning & chromatic burning classification of some graph family

Graph ‘G’ is a Simple and undirected graph, which has a lowest number of color that is required to color the edge is called chromatic index. It is denoted by the symbol χ1(G). In order to burn a graph, a minimum source of edges is required which is called edge burning index or edge burning number and it is denoted by the symbol b1(G). A minimum cardinality of color graph set and the source of the burning index set are called edge chromatic burning and let us assume that it is represented by the symbol χb1(G). In this paper we are mainly concentrating on identifying the chromatic burning and edge chromatic burning graphs and their chromatic burning index using a specific graph family. They are Petersen graph, Helm graph and n- Sunlet graph.

Local irregularity conjecture for 2-multigraphs versus cacti

A multigraph is locally irregular if the degrees of the end-vertices of every multiedge are distinct. The locally irregular coloring is an edge coloring of a multigraph \(G\) such that every color induces a locally irregular submultigraph of \(G\). A locally irregular colorable multigraph \(G\) is any multigraph which admits a locally irregular coloring. We denote by \(\textrm{lir}(G)\) the locally irregular chromatic index of a multigraph \(G\), which is the smallest number of colors required in the locally irregular coloring of the locally irregular colorable multigraph \(G\). In case of graphs the definitions are similar. The Local Irregularity Conjecture for 2-multigraphs claims that for every connected graph \(G\), which is not isomorphic to \(K_2\), multigraph \(^2G\) obtained from \(G\) by doubling each edge satisfies \(\textrm{lir}(^2G)\leq 2\). We show this conjecture for cacti. This class of graphs is important for the Local Irregularity Conjecture for 2-multigraphs and the Local Irregularity Conjecture which claims that every locally irregular colorable graph \(G\) satisfies \(\textrm{lir}(G)\leq 3\). At the beginning it has been observed that all not locally irregular colorable graphs are cacti. Recently it has been proved that there is only one cactus which requires 4 colors for a locally irregular coloring and therefore the Local Irregularity Conjecture was disproved.

Interval edge colorings of the generalized lexicographic product of some graphs

<p>An edge-coloring of a graph $ G $ with colors $ 1, \ldots, t $ is an interval $ t $-coloring if all colors are used and the colors of edges incident to each vertex of $ G $ are distinct and form an interval of integers. A graph $ G $ is interval colorable if it has an interval $ t $-coloring for some positive integer $ t $. For an interval colorable graph $ G $, the least and the greatest values of $ t $ for which $ G $ has an interval $ t $-coloring are denoted by $ w(G) $ and $ W(G) $. Let $ G $ be a graph with vertex set $ V(G) = \{u_1, \ldots, u_{m}\} $, $ m\geq2 $, and let $ h_m = (H_i)_{i\in\{1, \ldots, m\}} $ be a sequence of vertex-disjoint with $ V(H_i) = \{x_1^{(i)}, \ldots, x_{n_i}^{(i)}\} $, $ n_i\geq 1 $. The generalized lexicographic products $ G[h_m] $ of $ G $ and $ h_m $ is a simple graph with vertex set $ \cup_{i = 1}^{m}V(H_i) $, in which $ x_p^{(i)} $ is adjacent to $ x_q^{(j)} $ if and only if either $ u_i = u_j $ and $ x_p^{(i)}x_q^{(i)}\in E(H_i) $ or $ u_iu_j\in E(G) $. In this paper, we obtain several sufficient conditions for the generalized lexicographic product $ G[h_m] $ to have interval colorings. We also present some sharp bounds on $ w(G[h_m]) $ and $ W(G[h_m]) $ of $ G[h_m] $.</p>

PENERAPAN PEWARNAAN GRAF UNTUK MENINGKATKAN EFISIENSI PENJADWALAN MENGAJAR GURU DI MTS AL-KHOIRIYAH

Scheduling subjects in a school often gives a conflicting schedule to some teachers. On the other hand, graph theory can help to solve. One of the methods to solve the problem is implementing a coloring graph. In this paper, we apply the coloring graph using the Welch-Powell algorithm for scheduling subjects in MTs Al-Khoiriyah.Keywords: Welch-Powell algorithm, Coloring graph.

Local Irregularity Conjecture vs. cacti

A graph in which the two end-vertices of every edge have distinct degrees is called locally irregular. An edge coloring of a graph G such that every color induces a locally irregular subgraph of G is said to be locally irregular. A graph G which admits a locally irregular edge coloring is called colorable. The Local Irregularity Conjecture claims that 3 colors are enough for locally irregular edge coloring of any colorable graph. Not so long ago the only known graphs which do not admit a locally irregular 3-edge coloring were the non-colorable graphs, all of which are cacti. This fact motivated us to study locally irregular 3-edge colorings of cacti, which first resulted with the finding of the bow-tie graph B, a colorable cactus graph which requires 4 colors for a locally irregular edge coloring. As the graph B disproves the stated form of the Local Irregularity Conjecture, here we continue the study of locally irregular 3-edge colorings of cacti in relation to the conjecture. In this paper we confirm that all other cacti are locally irregular 3-edge colorable. This result makes us to believe that B is the only counterexample to the conjecture not only in the class of cacti, but also in general.

K4-intersecting families of graphs

Ellis, Filmus, and Friedgut proved an old conjecture of Simonovits and Sós showing that any triangle-intersecting family of graphs on n vertices has size at most 2(n2)−3, with equality for the family of graphs containing some fixed triangle. They conjectured that their results extend to cross-intersecting families, as well to Kt-intersecting families. We prove these conjectures for t∈{3,4}, showing that if F1 and F2 are families of graphs on n labeled vertices such that for any G1∈F1 and G2∈F2, G1∩G2 contains a Kt, then |F1||F2|≤4(n2)−(t2), with equality if and only if F1=F2 consists of all graphs that contain some fixed Kt. We also establish a stability result. More generally, “G1∩G2 contains a Kt” can be replaced by “G1 and G2 agree on a non-(t−1)-colorable graph.”

Decomposing graphs into interval colorable subgraphs and no-wait multi-stage schedules

A graph G is called interval colorable if it has a proper edge coloring with colors 1 , 2 , 3 , … such that the colors of the edges incident to every vertex of G form an interval of integers. Not all graphs are interval colorable; in fact, quite few families have been proved to admit interval colorings. In this paper we introduce and investigate a new notion, the interval coloring thickness of a graph G , denoted θ int ( G ) , which is the minimum number of interval colorable edge-disjoint subgraphs of G whose union is G . Our investigation is motivated by scheduling problems with compactness requirements, in particular, problems whose solution may consist of several schedules, but where each schedule must not contain any waiting periods or idle times for all involved parties. We first prove that every connected properly 3-edge colorable graph with maximum degree 3 is interval colorable, and using this result, we deduce an upper bound on θ int ( G ) for general graphs G . We demonstrate that this upper bound can be improved in the case when G is bipartite, planar or complete multipartite and consider some applications in timetabling.

Phenotypic characterization of donkey population in South Omo Zone, Southern Ethiopia

The study conducted in South Omo Zone, Southern Ethiopia with the aim of investigation within population diversification and characterization morphologically that helps to fill the gap of molecular characterization on this population. The data was collected from 500 adult donkeys from both sexes. Quantitative data was subjected to SAS GLM procedures by fitting districts and sex as the main effects. Qualitative data was subjected to a chi-square test with the district as the main effect. Color graph of donkey was done using Microsoft Excel, 2010. For both qualitative and quantitative data, the significance test was conducted at 5% of the level of error, and Tukey multiple range tests were used to separate the significance levels for the two types of data. CANDISC was used to calculate Mahalanobis distances, DISCRIM was used to cluster observations into predetermined groups, and STEPDISC was used to determine the quantitative characteristics that better differentiate populations. Roan coat color cover highest number compare with other coat color of donkey population. Quantitative traits of donkey has variation (P < 0.05) both in study areas and sex of donkeys. Overtly, except height at wither and height at the back Hammer donkey has mostly better metric value than the rest districts of the study areas. Moreover, CANDISC show variation on Hammer and Dasenech districts of donkey population. Furthermore, the longest (6.32) Mahalanobis distance observed in between Hammer and Dasenech donkey population. The Hammer and Dasenech donkey population is where the study fills in population variation the most. This can be because to management or genetics. Therefore, additional research might be required. Furthermore, morphometric measures show that donkey sex is similar, with the exception of heart girth circumference. This can be the result of poor selection, where superior male donkeys are sold for a higher price. Therefore, sound breeding programs should be used to reverse it.

Interval colorings of graphs—Coordinated and unstable no‐wait schedules

AbstractA proper edge‐coloring of a graph is an interval coloring if the labels on the edges incident to any vertex form an interval, that is, form a set of consecutive integers. The interval coloring thickness of a graph is the smallest number of interval colorable graphs edge‐decomposing . We prove that for any graph on vertices. This improves the previously known bound of , see Asratian, Casselgren, and Petrosyan. While we do not have a single example of a graph with an interval coloring thickness strictly greater than 2, we construct bipartite graphs whose interval coloring spectrum has arbitrarily many arbitrarily large gaps. Here, an interval coloring spectrum of a graph is the set of all integers such that the graph has an interval coloring using colors. Interval colorings of bipartite graphs naturally correspond to no‐wait schedules, say for parent–teacher conferences, where a conversation between any teacher and any parent lasts the same amount of time. Our results imply that any such conference with participants can be coordinated in no‐wait periods. In addition, we show that for any integers and , , there is a set of pairs of parents and teachers wanting to talk to each other, such that any no‐wait schedules are unstable—they could last hours and could last hours, but there is no possible no‐wait schedule lasting hours if .

Automated testing and interactive construction of unavoidable sets for graph classes of small path‐width

AbstractLet be a class of graphs with a membership test, , and let be the class of graphs in of path‐width at most . We present an interactive framework that finds an unavoidable set for , which is a set of graphs such that any graph in contains an isomorphic copy of a graph in . At the core of our framework is an algorithm that verifies whether a set of graphs is, indeed, unavoidable for . While obstruction sets are well‐studied, so far there is no general theory or algorithm for finding unavoidable sets. In general, it is undecidable whether a finite set of graphs is unavoidable for a given graph class. However, we give a criterion for termination: our algorithm terminates whenever is locally checkable of bounded maximum degree and is a finite set of connected graphs. For example, ‐regular graphs, ‐colourable graphs, and ‐free graphs are locally checkable classes. We put special emphasis on the case that is the class of cubic graphs and tailor the algorithm to this case. In particular, we introduce the new concept of high‐degree‐first path‐decompositions, which enables highly efficient pruning techniques. We exploit our framework to prove a new lower bound on the path‐width of cubic graphs. Moreover, we determine the extremal girth values of cubic graphs of path‐width for all and all smallest graphs which take on these extremal girth values. Further, we present a new constructive characterisation of the extremal cubic graphs of path‐width 3 and girth 4.

Terrain methods on spectral analysis for paleoclimate interpretations: A novel visualization technique using python

Spectral analysis techniques are valuable tools at the disposal of paleoclimate scientists in their research of cyclical phenomena potentially responsible for past climatic fluctuations. Advances in computing and an increased availability of climate time series have helped to consolidate this approach. Yet the visual representation of spectral analysis results has not improved at the same pace. Time-frequency analysis aims to identify periodic signals that vary over time using 2-D color graphs, depicting spectral bands theoretically discernible from the image background. The paleoclimate literature is full of examples such as the continuous wavelet analysis, the evolutionary fast Fourier transform (FFT) spectrogram, or even the more recent synchrosqueezing transform. Our approach is based on the stack of non-evolutive (assuming stationary behaviour) spectral analysis results from thousands of fixed interval time series, previously split from a longer and unevenly spaced (irregular sampling) paleoclimate time series. As illustrative examples, the targeted time series are derived from the LR04 Global Pliocene-Pleistocene Benthic δ18O stack and from the 65 °N summer insolation for the last 5.3 Myr, by means of the Lomb-Scargle periodogram technique. Enhanced and clearer visualization is achieved through the novel incorporation of the terrain analysis techniques: slope, hillshading and color mapping, and posterior blending of the individual images, using Python code. The result consists of a high resolution graphical output, allowing for better qualitative and quantitative interpretations of the obtained cyclicity, as the code incorporates the import of the achieved confidence levels from the spectral technique and the option to obscure the pixels under a certain threshold value. The application of terrain analysis techniques on visualization of spectral analysis results has the purpose of improving previous graphical representations of paleoclimate time-series, especially those time-frequency aspects of the involved phenomena. New developments of our approach may be applied to time-frequency analysis directly, supporting present and future paleoclimate studies.

Uniquely colorable graphs up to automorphisms

We extend the concept of uniquely colorable graphs and say that a graph G is χ-iso-unique if for every two proper colorings c:V(G)→{1,…,χ(G)} and d:V(G)→{1,…,χ(G)} there exists an automorphism φ of G such that for any i∈{1,…,χ(G)} there exists j∈{1,…,χ(G)} so that φ maps c−1(i) onto d−1(j). First, we prove some general properties about degrees of vertices of χ-iso-unique graphs G, and then focus on non-bipartite outerplanar χ-iso-unique graphs. As proved by Chartrand and Geller back in 1969, uniquely 3-colorable outerplanar graphs are precisely the 2-connected outerplanar graphs in which all induced cycles are triangles. We prove that a χ-iso-unique outerplanar graph G with χ(G)=3 contains at most one inner face, which is not a triangle, and it is isomorphic to C5 if it exists. Besides maximal outerplanar graphs, there is just one additional family of 2-connected outerplanar graphs that are χ-iso-unique, and they contain one facial 5-cycle. In contrast to the situation in uniquely colorable graphs, we find two families of outerplanar χ-iso-unique graphs having cut-vertices, and finally, we characterize all outerplanar χ-iso-unique graphs. We also show that χ-iso-unique outerplanar graphs can be recognized efficiently.

Global Christianity: A Guide to the World’s Largest Religion from Afghanistan to Zimbabwe, by Gina Zurlo

<i>Global Christianity: A Guide to the World’s Largest Religion from Afghanistan to Zimbabwe</i>, by Gina Zurlo

Topology and Adjunction in Promise Constraint Satisfaction

The approximate graph colouring problem, whose complexity is unresolved in most cases, concerns finding a $c$-colouring of a graph that is promised to be $k$-colourable, where $c\geq k$. This problem naturally generalises to promise graph homomorphism problems and further to promise constraint satisfaction problems. The complexity of these problems has recently been studied through an algebraic approach. In this paper, we introduce two new techniques to analyse the complexity of promise CSPs: one is based on topology and the other on adjunction. We apply these techniques, together with the previously introduced algebraic approach, to obtain new unconditional NP-hardness results for a significant class of approximate graph colouring and promise graph homomorphism problems.