Edge Coloring Research Articles

Extremal graphs for the distinguishing index

The distinguishing index D′(G) of a graph G is the least number of colours in an edge colouring, not necessarily proper, such that the identity is the only automorphism preserving it. This invariant is not defined for graphs having K2 as a component or more than one isolated vertex, and all other graphs we call admissible. In this paper we consider the following Turán-type problem: given an integer d≥2, what is a maximum size of an admissible graph G of order n such that D′(G)>d? The main result is the following theorem. If d≥2 and G is an admissible graph of order n≥d+4 and size‖G‖>(n−d−12)+d+1, then D′(G)≤d except for a few small graphs of order at most 7. We also exhibit all extremal graphs of order n≥9, that is, graphs G with ‖G‖=(n−d−12)+d+1 and D′(G)>d.

Avoiding and Extending Partial Edge Colorings of Hypercubes

We consider the problem of extending and avoiding partial edge colorings of hypercubes; that is, given a partial edge coloring varphi of the d-dimensional hypercube Q_d, we are interested in whether there is a proper d-edge coloring of Q_d that agrees with the coloring varphi on every edge that is colored under varphi ; or, similarly, if there is a proper d-edge coloring that disagrees with varphi on every edge that is colored under varphi . In particular, we prove that for any dge 1, if varphi is a partial d-edge coloring of Q_d, then varphi is avoidable if every color appears on at most d/8 edges and the coloring satisfies a relatively mild structural condition, or varphi is proper and every color appears on at most d-2 edges. We also show that varphi is avoidable if d is divisible by 3 and every color class of varphi is an induced matching. Moreover, for all 1 le k le d, we characterize for which configurations consisting of a partial coloring varphi of d-k edges and a partial coloring psi of k edges, there is an extension of varphi that avoids psi .

ON INTERVAL EDGE-COLORINGS OF COMPLETE MULTIPARTITE GRAPHS

A graph $G$ is called a complete $r$-partite $(r\geq 2)$ graph, if its vertices can be divided into $r$ non-empty independent sets $V_1,\ldots,V_r$ in a way that each vertex in $V_i$ is adjacent to all the other vertices in $V_j$ for $1\leq i<j\leq r$. Let $K_{n_{1},n_{2},\ldots,n_{r}}$ denote a complete $r$-partite graph with independent sets $V_1,V_2,\ldots,V_r$ of sizes $n_{1},n_{2},\ldots,n_{r}$. An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ is called an \emph{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. In this paper we have obtained some results on the existence and construction of interval edge-colorings of complete $r$-partite graphs. Moreover, we have also derived an upper bound on the number of colors in interval colorings of complete multipartite graphs.

Ramsey Numbers of Fans and Large Books

For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum integer $N$ such that any red/blue edge coloring of $K_N$ contains either a red $G$ or a blue $H$. Let $G+H$ be the graph obtained from vertex disjoint $G$ and $H$ by adding new edges connecting $G$ and $H$ completely, $F_m=K_1+mK_2$ and $B_p(n)=K_p+nK_1$. It is shown $R(F_m,B_p(n))=2(n+p-1)+1$ for fixed $m, p$ and large $n$.

RAINBOW CONNECTION NUMBER AND TOTAL RAINBOW CONNECTION NUMBER OF AMALGAMATION RESULTS DIAMOND GRAPH(〖Br〗_4) AND FAN GRAPH(F_3)

If be a graph and edge coloring of G is a function , rainbow connection number is the minimum-k coloration of the rainbow on the edge of graph G and denoted by rc(G). Rainbow connection numbers can be applied to the result of operations on some special graphs, such as diamond graphs and fan graphs. Graph operation is a method used to obtain a new graph by combining two graphs. This study performed amalgamation operations to obtain rainbow connection numbers and rainbow-total-connection numbers in diamond graphs ( ) and fan graphs ( ) or . Based on the research, it is obtained that the rainbow-connection number theorem on the amalgamation result of the diamond graph ( ) and fan graph ( is with . Furthermore, the theorem related to the total rainbow-connection number on the amalgamation result of the diamond graph( ) and the fan graph ( is obtained, namely with .

On twin edge colorings in m-ary trees

Let k ≥ 2 be an integer and G be a connected graph of order at least 3 . A twin k -edge coloring of G is a proper edge coloring of G that uses colors from ℤ k and that induces a proper vertex coloring on G where the color of a vertex v is the sum (in ℤ k ) of the colors of the edges incident with v . The smallest integer k for which G has a twin k -edge coloring is the twin chromatic index of G and is denoted by χ ′ t ( G ) . In this paper, we study the twin edge colorings in m -ary trees for m ≥ 2 ; in particular, the twin chromatic indexes of full m -ary trees that are not stars, r -regular trees for even r ≥ 2 , and generalized star graphs that are not paths nor stars are completely determined. Moreover, our results confirm the conjecture that χ ′ t ( G )≤ Δ ( G )+2 for every connected graph G (except C 5 ) of order at least 3 , for all trees of order at least 3 .

Grünbaum colorings extended to non-facial 3-cycles

We consider the question of when a triangulation with a Grünbaum coloring can be edge-colored with three colors such that the non-facial 3-cycles also receive all three colors; we will call this a strong Grünbaum coloring . It turns out that for the sphere, every triangulation has a strong Grünbaum coloring, and that the presence of a K 5 subgraph prohibits a strong Grünbaum coloring, but that K 5 is not the only such barrier. We investigate the ramifications of these facts. We also show that for every other topological surface there exist triangulations with a strong Grünbaum coloring and triangulations that have Grünbaum colorings but that cannot have a strong Grünbaum coloring. Finally, we reframe strong Grünbaum colorings as certain hypergraph edge colorings, and raise the question of how many colors are needed to achieve an edge coloring such that both facial and non-facial 3-cycles receive three colors.

The strong 3-rainbow index of edge-comb product of a path and a connected graph

Let G be a connected and edge-colored graph of order n , where adjacent edges may be colored the same. A tree in G is a rainbow tree if all of its edges have distinct colors. Let k be an integer with 2 ≤ k ≤ n . The minimum number of colors needed in an edge coloring of G such that there exists a rainbow tree connecting S with minimum size for every k -subset S of V ( G ) is called the strong k -rainbow index of G , denoted by s r x k ( G ) . In this paper, we study the s r x 3 of edge-comb product of a path and a connected graph, denoted by P n o ⊳ e H . It is clearly that | E ( P n o ⊳ e H )| is the trivial upper bound for s r x 3 ( P n o ⊳ e H ) . Therefore, in this paper, we first characterize connected graphs H with s r x 3 ( P n o ⊳ e H )=| E ( P n o ⊳ e H )| , then provide a sharp upper bound for s r x 3 ( P n o ⊳ e H ) where s r x 3 ( P n o ⊳ e H )≠| E ( P n o ⊳ e H )| . We also provide the exact value of s r x 3 ( P n o ⊳ e H ) for some connected graphs H .

Hidden transition in multiplex networks

Weak multiplex percolation generalizes percolation to multi-layer networks, represented as networks with a common set of nodes linked by multiple types (colors) of edges. We report a novel discontinuous phase transition in this problem. This anomalous transition occurs in networks of three or more layers without unconnected nodes, P(0),=,0. Above a critical value of a control parameter, the removal of a tiny fraction Delta of nodes or edges triggers a failure cascade which ends either with the total collapse of the network, or a return to stability with the system essentially intact. The discontinuity is not accompanied by any singularity of the giant component, in contrast to the discontinuous hybrid transition which usually appears in such problems. The control parameter is the fraction of nodes in each layer with a single connection, Pi ,=,P(1). We obtain asymptotic expressions for the collapse time and relaxation time, above and below the critical point Pi _c, respectively. In the limit Delta rightarrow 0 the total collapse for Pi ,>,Pi _text {c} takes a time T propto 1/(Pi -Pi _text {c}), while there is an exponential relaxation below Pi _text {c} with a relaxation time tau propto 1/[Pi _text {c}-Pi ].

LineGAN: An image colourisation method combined with a line art network

The work on grayscale image colourisation has been significantly improved. Currently, learning-based methods have achieved some great colourisation effects, but existing colour edge bleeding, especially when colourful cartoon characters. In this paper, we focus on the colourisation of cartoon characters from a series in an adversarial environment with a line art network, whose name is LineGAN. LineGAN learns the corresponding colour mapping from datasets, improving the accuracy of image colourisation. Our methods limit the colour boundary overflow by adding a line art frame in the generator. Extensive experiment results on cartoon image colourisation tasks demonstrate that the proposed method can achieve effective results.

Color segmentation of multicolor porous printed fabrics by conjugating SOM and EDSC clustering algorithms

In order to achieve accurate color segmentation of printed fabrics, the color segmentation algorithm combining the self-organizing maps neural network and the efficient dense subspace clustering was proposed in this paper. After pre-processing of the fabric image, the primary clustering was implemented by the self-organizing maps algorithm, then the secondary clustering was done by the efficient dense subspace clustering algorithm. The optimal silhouette coefficient is introduced into the clustering process of the efficient dense subspace clustering algorithm to determine the number of clustering centers automatically. Finally, by the post-processing including gray-scale transformation, binarization and open operation, the mis-segmentation of edge color was eliminated, making the algorithm more suitable for industrial application. Experiments were carried out and results show that the algorithm proposed in this paper can recognize the color of small areas accurately and segment the color of complex printed fabric images. The color segmented results of 20 printed fabrics show that the accuracy of the algorithm proposed in this paper reaches 88.3%.

Study of algorithm for coloring in various graph

Graph coloring is an important area of mathematics and computer science. Graph coloring problem is getting more famous to solve the variety of real-world problems like map coloring, timetabling and scheduling. Graph coloring is allied with two types of coloring as vertex and edge coloring. Algorithm is a set of rules that must be followed when solving a particular coloring problem and algorithms plays an important role in graph coloring. The main objective of this paper is to study of various algorithms in graph coloring.

Strong Edge Coloring of Cayley Graphs and Some Product Graphs

A strong edge coloring of a graph G is a proper edge coloring of G such that every color class is an induced matching. The minimum number of colors required is termed the strong chromatic index. In this paper we determine the exact value of the strong chromatic index of all unitary Cayley graphs. Our investigations reveal an underlying product structure from which the unitary Cayley graphs emerge. We then go on to give tight bounds for the strong chromatic index of the Cartesian product of two trees, including an exact formula for the product in the case of stars. Further, we give bounds for the strong chromatic index of the product of a tree with a cycle. For any tree, those bounds may differ from the actual value only by not more than a small additive constant (at most 2 for even cycles and at most 4 for odd cycles), moreover they yield the exact value when the length of the cycle is divisible by 4.

Strong edge colorings of graphs and the covers of Kneser graphs

AbstractA proper edge coloring of a graph is strong if it creates no bichromatic path of length three. It is well known that for a strong edge coloring of a ‐regular graph at least colors are needed. We show that a ‐regular graph admits a strong edge coloring with colors if and only if it covers the Kneser graph . In particular, a cubic graph is strongly 5‐edge‐colorable whenever it covers the Petersen graph. One of the implications of this result is that a conjecture about strong edge colorings of subcubic graphs proposed by Faudree et al. is false.

Pathological characterization and cause of a novel liver disease in largemouth bass (Micropterus salmoides)

With the rapid development of intensive aquaculture in recent years, liver disease of largemouth bass has been increasingly prominent both in China and even around the world. The main gross pathology found the liver is swollen, pale or white in color, with round and blunt edges, and brittle texture. The morbidity and mortality varied. In some cases, the cumulative mortality could reach more than 20%. To investigate the etiology of largemouth bass, histopathological and ultrastructural pathology were performed. The results showed that diseased fish expressed fat accumulation around the pyloric caeca, and the liver showed hepatomegaly with extensive yellowish or whitish color. Hepatocytes had degenerated vacuoles that were different from fatty liver disease. Large number of irregular vesicles that accumulated in hepatic cells which contained a mass of glycogen-like granules and myeloid bodies. This novel liver disease was a common phenomenon in the culture of largemouth bass. The animal feeding experiment was to investigate the causes of the disease, and the results confirmed that a high carbohydrate diet was one of the causes of novel liver disease of largemouth bass. Additionally, blood biochemical indexes suggested that the disease can cause systemic metabolic disorder. Overall, the results demonstrated that the death of a large number of largemouth bass was caused by novel liver disease, due to a high carbohydrate diet.

Morphological and molecular identification of multi-antibiotic resistant bacteria in the wound site of diabetic ulcers

Abstract. Risqiyah W, Narulita E, Rofiqoh A, Ludfi AS, Iqbal M. 2021. Morphological and molecular identification of multi-antibiotic resistant bacteria in the wound site of diabetic ulcers. Biodiversitas 23: 663-670. The study aimed to determine the characteristics of diverse species of bacteria found in diabetic ulcers and the inhibition of antibiotics against these bacteria. The method used is Gram staining for morphological characterization, identification using 16S rRNA gene, and pour plate for antibiotic resistance test. The morphological characterization result showed that the colonies had white color, circular shape, flat elevation, entire edge, small size, and basil shape with Gram negative type of bacteria. Another result showed yellow, greenish, pellucid, and green color, irregular shape, raised elevation, undulate and lobate edge, medium size, coccus and Gram positive type of bacteria. Query cover of molecular identification showed 73-100% and 77.61-96.77% for the result of similarity identification with Alcaligenes faecalis, Shigella flexneri, Enterococcus faecalis, Proteus mirabilis, and Acinetobacter seohaensis. The identification and antibiotic resistance testing showed that the bacterial species found in diabetic ulcers was A. faecalis strain NRBC 13111, which was resistant to all tested antibiotics, except A. faecalis strain NRBC 13111 from the UB 2.3K and A. faecalis strain NRBC 13111 from the sample UB 3.4M. Enterococcus faecalis strain ATCC 19433 was resistant to ceftriaxone, ceftazidime, clindamycin and metronidazole, intermediates to cefoperazone, and sensitive to ciprofloxacin. Proteus mirabilis strain JCM 1669 was resistant to clindamycin and metronidazole, sensitive to ceftriaxone and ciprofloxacin, intermediates to ceftazidime and cefoperazone. Proteus mirabilis strain ATCC 29906, A. seohaensis strain SW-100 and S. flexneri strain ATCC 29903 were resistant to all tested antibiotics.

Graph coloring using the reduced quantum genetic algorithm.

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.

Edge homogeneous colorings

We explore four kinds of edge colorings defined by the requirement of equal number of colors appearing, in particular ways, around each vertex or each edge. We obtain the characterization of graphs colorable in such a way that the ends of each edge see (not regarding the edge color itself) \(q\) colors (resp. one end sees \(q\) colors and the color sets for both ends are the same), and a sufficient condition for 2-coloring a graph in a way that the ends of each edge see (with the omission of that edge color) altogether \(q\) colors. The relations of these colorings to \(M_q\)-colorings and role colorings are also discussed; we prove an interpolation theorem for the numbers of colors in edge coloring where all edges around each vertex have \(q\) colors.

On Rainbow Cycles and Proper Edge Colorings of Generalized Polygons

An edge coloring of a simple graph G is said to be proper rainbow-cycle-forbidding (PRCF, for short) if no two incident edges receive the same color and for any cycle in G, at least two edges of that cycle receive the same color. A graph G is defined to be PRCF-good if it admits a PRCF edge coloring, and G is deemed PRCF-bad otherwise. In recent work, Hoffman, et al. study PRCF edge colorings and find many examples of PRCF-bad graphs having girth less than or equal to 4. They then ask whether such graphs exist having girth greater than 4. In our work, we give a straightforward counting argument showing that the Hoffman-Singleton graph answers this question in the affirmative for the case of girth 5. It is then shown that certain generalized polygons, constructed of sufficiently large order, are also PRCF-bad, thus proving the existence of PRCF-bad graphs of girth 6, 8, 12, and 16.

Rainbow Perfect and Near-Perfect Matchings in Complete Graphs with Edges Colored by Circular Distance

Given an edge-colored complete graph Kn on n vertices, a perfect (respectively, near-perfect) matching M in Kn with an even (respectively, odd) number of vertices is rainbow if all edges have distinct colors. In this paper, we consider an edge coloring of Kn by circular distance, and we denote the resulting complete graph by K●n. We show that when K●n has an even number of vertices, it contains a rainbow perfect matching if and only if n=8k or n=8k+2, where k is a nonnegative integer. In the case of an odd number of vertices, Kirkman matching is known to be a rainbow near-perfect matching in K●n. However, real-world applications sometimes require multiple rainbow near-perfect matchings. We propose a method for using a recursive algorithm to generate multiple rainbow near-perfect matchings in K●n.