Chromatic Number Of Graph Research Articles

Coloring [formula omitted]-partite sparse digraphs

In this paper, we study colorings of k-partite sparse digraphs. The chromatic number of a graph G is the smallest integer k such that the vertices of G can be colored with k colors with the property that each color class is an independent set. The dichromatic number of a digraph D is the minimum k such that the vertices of D can be colored with k colors with each color class inducing an acyclic subdigraph. This coloring invariant shares many similarities with the graph chromatic number and can be thought of as its analogous digraph generalization.Our main result in this short note shows that there exist sparse k-partite digraphs which have dichromatic number k. This, in particular, not only implies that there exist graphs with equal chromatic and dichromatic number, but that they can be taken to be somewhat sparse.

Neighbor Distinguishing Colorings of Graphs with the Restriction for Maximum Average Degree

Neighbor distinguishing colorings of graphs represent powerful tools for solving the channel assignment problem in wireless communication networks. They consist of two forms of coloring: neighbor distinguishing edge coloring, and neighbor distinguishing total coloring. The neighbor distinguishing edge (total) coloring of a graph G is an edge (total) coloring with the requirement that each pair of adjacent vertices contains different color sets. The neighbor distinguishing edge (total) chromatic number of G is the smallest integer k in cases where a neighbor distinguishing edge (total) coloring exists through the use of k colors in G. The maximum average degree of G is the maximum of the average degree of its non-empty subgraphs. In this paper, we characterize the neighbor distinguishing edge (total) chromatic numbers of graphs with a maximum average degree less than four by means of the discharging method.

Observations on the Lovász θ-Function, Graph Capacity, Eigenvalues, and Strong Products †.

This paper provides new observations on the Lovász θ-function of graphs. These include a simple closed-form expression of that function for all strongly regular graphs, together with upper and lower bounds on that function for all regular graphs. These bounds are expressed in terms of the second-largest and smallest eigenvalues of the adjacency matrix of the regular graph, together with sufficient conditions for equalities (the upper bound is due to Lovász, followed by a new sufficient condition for its tightness). These results are shown to be useful in many ways, leading to the determination of the exact value of the Shannon capacity of various graphs, eigenvalue inequalities, and bounds on the clique and chromatic numbers of graphs. Since the Lovász θ-function factorizes for the strong product of graphs, the results are also particularly useful for parameters of strong products or strong powers of graphs. Bounds on the smallest and second-largest eigenvalues of strong products of regular graphs are consequently derived, expressed as functions of the Lovász θ-function (or the smallest eigenvalue) of each factor. The resulting lower bound on the second-largest eigenvalue of a k-fold strong power of a regular graph is compared to the Alon-Boppana bound; under a certain condition, the new bound is superior in its exponential growth rate (in k). Lower bounds on the chromatic number of strong products of graphs are expressed in terms of the order and the Lovász θ-function of each factor. The utility of these bounds is exemplified, leading in some cases to an exact determination of the chromatic numbers of strong products or strong powers of graphs. The present research paper is aimed to have tutorial value as well.

Graph clustering via generalized colorings

We propose a new approach for defining and searching clusters in graphs that represent real technological or transaction networks. In contrast to the standard way of finding dense parts of a graph, we concentrate on the structure of edges between the clusters, as it is motivated by some earlier observations, e.g. in the structure of networks in ecology and economics and by applications of discrete tomography. Mathematically special colorings and chromatic numbers of graphs are studied.

Implementation of Graph Theory on the Setting of the Traffic Light Intersection

On the crossroads of course required setting traffic lights for vehicles that can be flashed with a sense of security and comfort. The duration of the traffic lights can be set to the maximum extent possible so that the vehicle could pass the road regularly. There are many solutions to manage traffic at a crossroads. One of them is a solution which will be given through this observation. This paper aims to describe the related optimization settings traffic by using the algorithm staining which aims to get the chromatic number of graphs. The optimization needs to be done due to the settings of the traffic at the Intersection of Palimanan often seen traffic jams and crowded at rush hour as a result of policies that allow the current to be incompatible. The application of the algorithm staining Welch – Powel to determine the chromatic number of graphs can be another option to resolve the problem. After the current – the current at the traffic being transformed into a graph model that will be coloring a graph and calculating the number lens. Chromatic number of the graph model is 4 so there is a change in the duration of the red and green lights on the traffic. Although there is a reduction in the duration of the red light and the addition of the duration of the green light, the removal of the current incompatibility can minimize the possibility of congestion.

On the smallest signless Laplacian eigenvalue of graphs

For a simple graph G the signless Laplacian matrix of G is defined as D(G)+A(G), where A(G) and D(G) are the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. By the smallest signless Laplacian eigenvalue of G, denoted by q′(G), we mean the smallest eigenvalue of the signless Laplacian matrix of G. In this paper we study the smallest signless Laplacian eigenvalue of graphs and find some relations between this and the chromatic number of graphs. We prove that if G is a graph of order n and with chromatic number χ(G), then q′(G)≤q′(T(n,χ(G))), where T(n,t) is the Turán graph on n vertices and t parts. Using this inequality we obtain some bounds for q′(G) that improve the known previous bounds.

A New Upper Bound on the Chromatic Number of Graphs with No Odd Kt Minor

Gerards and Seymour conjectured that every graph with no odd Kt minor is (t − 1)-colorable. This is a strengthening of the famous Hadwiger’s Conjecture. Geelen et al. proved that every graph with no odd Kt minor is \(O(t\sqrt {\log t} )\)-colorable. Using the methods the present authors and Postle recently developed for coloring graphs with no Kt minor, we make the first improvement on this bound by showing that every graph with no odd Kt minor is O(t(logt)β)-colorable for every β > 1/4.

Clique minors in graphs with a forbidden subgraph

AbstractThe classical Hadwiger conjecture dating back to 1940s states that any graph of chromatic number at least r has the clique of order r as a minor. Hadwiger's conjecture is an example of a well‐studied class of problems asking how large a clique minor one can guarantee in a graph with certain restrictions. One problem of this type asks what is the largest size of a clique minor in a graph on n vertices of independence number at most r. If true Hadwiger's conjecture would imply the existence of a clique minor of order . Results of Kühn and Osthus and Krivelevich and Sudakov imply that if one assumes in addition that G is H‐free for some bipartite graph H then one can find a polynomially larger clique minor. This has recently been extended to triangle‐free graphs by Dvořák and Yepremyan, answering a question of Norin. We complete the picture and show that the same is true for arbitrary graph H, answering a question of Dvořák and Yepremyan. In particular, we show that any ‐free graph has a clique minor of order , for some constant depending only on s. The exponent in this result is tight up to a constant factor in front of the term.

On r-dynamic chromatic number of some brick product graphs C(2n, 1, p)

An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that [Formula: see text] for each vertex [Formula: see text]. The r-dynamic chromatic number of a graph G is the minimum k such that G has an r-dynamic coloring with k colors. In this paper, we obtain the r-dynamic chromatic number of brick product graphs.

On Bi-gram Graph Attributes

We propose a new approach to text semantic analysis and general corpus analysis using, as termed in this article, a "bi-gram graph" representation of a corpus. The different attributes derived from graph theory are measured and analyzed as unique insights or against other corpus graphs, attributes such as the graph chromatic number and the graph coloring, graph density and graph K-core. We observe a vast domain of tools and algorithms that can be developed on top of the graph representation; creating such a graph proves to be computationally cheap, and much of the heavy lifting is achieved via basic graph calculations. Furthermore, we showcase the different use-cases for the bi-gram graphs and how scalable it proves to be when dealing with large datasets.

On d-distance equitable chromatic number of some graphs

An assignment of distinct colors [Formula: see text] to the vertices [Formula: see text] and [Formula: see text] of a graph [Formula: see text] such that the distance between [Formula: see text] and [Formula: see text] is at most [Formula: see text] is called [Formula: see text]-distance coloring of [Formula: see text]. Suppose [Formula: see text] are the color classes of [Formula: see text]-distance coloring and [Formula: see text] for any [Formula: see text], then [Formula: see text] is [Formula: see text]-distance equitable colored graph. In this paper, we obtain [Formula: see text]-distance chromatic number and [Formula: see text]-distance equitable chromatic number of graphs like [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].

Structure and Colour in Triangle-Free Graphs

Motivated by a recent conjecture of the first author, we prove that every properly coloured triangle-free graph of chromatic number $\chi$ contains a rainbow independent set of size $\lceil\frac12\chi\rceil$. This is sharp up to a factor $2$. This result and its short proof have implications for the related notion of chromatic discrepancy.
 Drawing inspiration from both structural and extremal graph theory, we conjecture that every triangle-free graph of chromatic number $\chi$ contains an induced cycle of length $\Omega(\chi\log\chi)$ as $\chi\to\infty$. Even if one only demands an induced path of length $\Omega(\chi\log\chi)$, the conclusion would be sharp up to a constant multiple. We prove it for regular girth $5$ graphs and for girth $21$ graphs.
 As a common strengthening of the induced paths form of this conjecture and of Johansson's theorem (1996), we posit the existence of some $c >0$ such that for every forest $H$ on $D$ vertices, every triangle-free and induced $H$-free graph has chromatic number at most $c D/\log D$. We prove this assertion with 'triangle-free' replaced by 'regular girth 5'.

On distance Laplacian spectral radius and chromatic number of graphs

Let G be a connected simple graph with n vertices having chromatic number χ. The distance Laplacian matrix DL(G) is defined as DL(G)=Diag(Tr)−D, where Diag(Tr) is the diagonal matrix of vertex transmissions and D is the distance matrix of G. The eigenvalues of DL(G) are the distance Laplacian eigenvalues of G and are denoted by ∂1L(G),∂2L(G),…,∂nL(G). The largest eigenvalue ∂1L(G) is called the distance Laplacian spectral radius. For a non-complete graph G with n vertices and chromatic number χ, Aouchiche and Hansen (2017) proved that ∂1L(G)≥n+⌈nχ⌉. If G is a connected graph with n≥4 vertices and chromatic number χ≤n−2, we prove that ∂2L(G)≥n+⌈nχ⌉ and we show the existence of graphs for which the equality holds. Among all graphs with chromatic number χ satisfying n2≤χ≤n−1, we show that the graph K2,2,…,2︸n−χ,1,1,…,1︸2χ−n has the minimum distance Laplacian spectral radius. Also, we give the distribution of the distance Laplacian eigenvalues in relation to the chromatic number χ and other graph invariants. We characterize the extremal graphs for some of these results and for others, we illustrate by examples.

A study of the neighborhood complex of $-stable Kneser graphs

In 1978, Alexander Schrijver defined the stable Kneser graphs as a vertex critical subgraphs of the Kneser graphs. In the early 2000s, Günter M. Ziegler generalized Schrijver’s construction and defined the s-stable Kneser graphs. Thereafter Frédéric Meunier determined the chromatic number of the s-stable Kneser graphs for special cases and formulated a conjecture on the chromatic number of the s-stable Kneser graphs. In this paper we study a generalization of the s-stable Kneser graphs. For some specific values of the parameter we show that the neighborhood complex of < s, t >-stable Kneser graph has the same homotopy type as the (t − 1)-sphere. In particular, this implies that the chromatic number of this graph is t + 1.

Topological Bounds for Graph Representations over Any Field

Haviv [European J. Combin., 81 (2019), pp. 84--97] has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension ov...

The Chromatic Number of Graphs with No Induced Subdivision of $$K_4$$

In 2012, Leveque, Maffray and Trotignon conjectured that if a graph does not contain an induced subdivision of $$K_4$$, then it is 4-colorable. Recently, Le showed that every such graph is 24-colorable. In this paper, we improve the upper bound to 8.

On the chromatic number of disjointness graphs of curves

Let ω(G) and χ(G) denote the clique number and chromatic number of a graph G, respectively. The disjointness graph of a family of curves (continuous arcs in the plane) is the graph whose vertices correspond to the curves and in which two vertices are joined by an edge if and only if the corresponding curves are disjoint. A curve is called x-monotone if every vertical line intersects it in at most one point. An x-monotone curve is grounded if its left endpoint lies on the y-axis.We prove that if G is the disjointness graph of a family of grounded x-monotone curves such that ω(G)=k, then χ(G)≤(k+12). If we only require that every curve is x-monotone and intersects the y-axis, then we have χ(G)≤k+12(k+23). Both of these bounds are best possible. The construction showing the tightness of the last result settles a 25 years old problem: it yields that there exist Kk-free disjointness graphs of x-monotone curves such that any proper coloring of them uses at least Ω(k4) colors. This matches the upper bound up to a constant factor.

О ХРОМАТИЧЕСКОМ ЧИСЛЕ ГРАФОВ С НЕКОТОРЫМ ОГРАНИЧЕНИЕМ СТЕПЕНЕЙ ВЕРШИН

О ХРОМАТИЧЕСКОМ ЧИСЛЕ ГРАФОВ С НЕКОТОРЫМ ОГРАНИЧЕНИЕМ СТЕПЕНЕЙ ВЕРШИН

关于图的模染色数的上界

关于图的模染色数的上界

Matching Polytons

Hladký, Hu, and Piguet [Tilings in graphons, preprint] introduced the notions of matching and fractional vertex covers in graphons. These are counterparts to the corresponding notions in finite graphs. 
 Combinatorial optimization studies the structure of the matching polytope and the fractional vertex cover polytope of a graph. Here, in analogy, we initiate the study of the structure of the set of all matchings and of all fractional vertex covers in a graphon. We call these sets the matching polyton and the fractional vertex cover polyton.
 We also study properties of matching polytons and fractional vertex cover polytons along convergent sequences of graphons.
 As an auxiliary tool of independent interest, we prove that a graphon is $r$-partite if and only if it contains no graph of chromatic number $r+1$. This in turn gives a characterization of bipartite graphons as those having a symmetric spectrum.