Mycielski Graphs Research Articles

Roman Domination in Mycielski Graphs: A Study of Some Graphs and a Heuristic Algorithm

Let G=(V,E) be a graph. A function f:V→\{0,1,2\}, if ∀u for which f(u)=0 is adjacent to ∃v for which f(v)=2, is called a Roman dominating function, and called in short terms RDF. The weight of an RDF f is f(V)=∑_(v∈V)▒f(v) . The Roman domination number of a graph G, denoted by γ_R (G), is the minimum weight of an RDF on G. This paper presents the results for Roman domination numbers of the Mycielski graphs obtained through Mycielski's construction of the comet, double comet, and comb graphs. An algorithm to determine the Roman domination number of any given graph is also provided.

On Geodesic Convexity in Mycielskian of Graphs

The convexity induced by the geodesics in a graph G is called the geodesic convexity of G. Mycielski graphs preserve the property of being triangle-free and many parameters such as power domination number, coloring number, determining number and recently general position number have been determined for them. In this work, we determine the geodesic convexity parameters viz., convexity, geodetic iteration, geodetic, and hull numbers for Mycielski graphs for which the underlying graphs considered are path, cycle, star, and complete graph.

On equitable near-proper coloring of some derived graph classes

An equitable near-proper coloring of a graph $G$ is a defective coloring in which the number of vertices in any two color classes differ by at most one and the bad edges obtained is minimised by restricting the number of color classes that can have adjacency among their own elements. This paper investigates the equitable near-proper coloring of some derived graph classes like Mycielski graphs, splitting graphs and shadow graphs.

$L(2,1)$-labeling of the iterated Mycielski graphs of graphs and some problems related to matching problems

$L(2,1)$-labeling of the iterated Mycielski graphs of graphs and some problems related to matching problems

On multichromatic numbers of widely colorable graphs

AbstractA coloring is called ‐wide if no walk of length connects vertices of the same color. A graph is ‐widely colorable with colors if and only if it admits a homomorphism into a universal graph . Tardif observed that the value of the multichromatic number of these graphs is at least and equality holds for . He asked whether there is equality also for . We show that for all thereby answering Tardif's question. We observe that for large (with respect to and fixed) we cannot have equality and that for fixed and going to infinity the fractional chromatic number of also tends to infinity. The latter is a simple consequence of another result of Tardif on the fractional chromatic number of generalized Mycielski graphs.

On equitable chromatic topological indices of some Mycielski graphs

In recent years, the notion of chromatic Zagreb indices has been introduced and studied for certain basic graph classes, as a coloring parallel of different Zagreb indices. A proper coloring C of a graph G, which assigns colors to the vertices of G such that the numbers of vertices in any two color classes differ by at most one, is called an equitable coloring of G. In this paper, we introduce the equitable chromatic Zagreb indices and equitable chromatic irregularity indices of some special classes of graphs called Mycielski graphs of paths and cycles.

Signed domination and Mycielski’s structure in graphs

LetG = (V, E) be a graph. The functionf : V(G) → {−1, 1} is a signed dominating function if for every vertexv ∈ V(G), ∑x∈NG[v]f(x)≥1. The value ofω(f) = ∑x∈V(G)f(x) is called the weight off. The signed domination number ofGis the minimum weight of a signed dominating function ofG. In this paper, we initiate the study of the signed domination numbers of Mycielski graphs and find some upper bounds for this parameter. We also calculate the signed domination number of the Mycielski graph when the underlying graph is a star, a wheel, a fan, a Dutch windmill, a cycle, a path or a complete bipartite graph.

Reconfiguring graph homomorphisms on the sphere

Given a loop-free graph H, the reconfiguration problem for homomorphisms to H (also called H-colourings) asks: given two H-colourings f of g of a graph G, is it possible to transform f into g by a sequence of single-vertex colour changes such that every intermediate mapping is an H-colouring? This problem is known to be polynomial-time solvable for a wide variety of graphs H (e.g. all C4-free graphs) but only a handful of hard cases are known. We prove that this problem is PSPACE-complete whenever H is a K2,3-free quadrangulation of the 2-sphere (equivalently, the plane) which is not a 4-cycle. From this result, we deduce an analogous statement for non-bipartite K2,3-free quadrangulations of the projective plane. This include several interesting classes of graphs, such as odd wheels, for which the complexity was known, and 4-chromatic generalized Mycielski graphs, for which it was not.If we instead consider graphs G and H with loops on every vertex (i.e. reflexive graphs), then the reconfiguration problem is defined in a similar way except that a vertex can only change its colour to a neighbour of its current colour. In this setting, we use similar ideas to show that the reconfiguration problem for H-colourings is PSPACE-complete whenever H is a reflexive K4-free triangulation of the 2-sphere which is not a reflexive triangle. This proof applies more generally to reflexive graphs which, roughly speaking, resemble a triangulation locally around a particular vertex. This provides the first graphs for which the reconfiguration problem is known to be PSPACE-complete for reflexive instances.

Strong total domination and weak total domination in Mycielski's graphs

Strong total domination and weak total domination in Mycielski's graphs

Generalised Mycielski Graphs and the Borsuk–Ulam Theorem

Stiebitz determined the chromatic number of generalised Mycielski graphs using the topological method of Lovász, which invokes the Borsuk–Ulam theorem. Van Ngoc and Tuza used elementary combinatorial arguments to prove Stiebitz's theorem for 4-chromatic generalised Mycielski graphs, and asked if there is also an elementary combinatorial proof for higher chromatic number. We answer their question by showing that Stiebitz's theorem can be deduced from a version of Fan's combinatorial lemma. Our proof uses topological terminology, but is otherwise completely discrete and could be rewritten to avoid topology altogether. However, doing so would be somewhat artificial, because we also show that Stiebitz's theorem is equivalent to the Borsuk–Ulam theorem.

On atom-bond connectivity index of graphs

The atom-bond connectivity index (ABC-index) is a useful topological index employed in studying the stability of alkanes and the strain energy of cycloalkanes. The ABC-index of a nontrivial graph G, denoted by ABC(G), is defined as ABC(G)=∑vivj∈E(G)di+dj−2didj, where di is the degree of vertex vi in G. In this paper, we first present some lower bounds for ABC-index by means of some known inequalities. Then we give some upper bounds for ABC-index in terms of graph parameters such as clique number, vertex connectivity, algebraic connectivity and spectral radius. Moreover, we give some lower and upper bounds for ABC-index of Mycielski graphs. Finally, we compare ABC-index with other graph invariants for connected graphs.

Neighbor-locating colorings in graphs

A k-coloring of a graph G is a k-partition Π={S1,…,Sk} of V(G) into independent sets, called colors. A k-coloring is called neighbor-locating if for every pair of vertices u,v belonging to the same color Si, the set of colors of the neighborhood of u is different from the set of colors of the neighborhood of v. The neighbor-locating chromatic numberχNL(G) is the minimum cardinality of a neighbor-locating coloring of G.We establish some tight bounds for the neighbor-locating chromatic number of a graph, in terms of its order, maximum degree and independence number. We determine all connected graphs of order n≥5 with neighbor-locating chromatic number n or n−1. We examine the neighbor-locating chromatic number for two graph operations: join and disjoint union, and also for two graph families: split graphs and Mycielski graphs.

Bounds for the boxicity of Mycielski graphs

A box in Euclidean k-space is the Cartesian product of k closed intervals on the real line. The boxicity of a graph G, denoted by box(G), is the minimum nonnegative integer k such that G can be isomorphic to the intersection graph of a family of boxes in Euclidean k-space. Mycielski introduced an interesting graph operation that extends a graph G to a new graph M(G), called the Mycielski graph of G. In this paper we observe the behavior of boxicity of Mycielski graphs. We see that box(M(G)) is at least box(G) for a graph G, and hence we are interested in whether the boxicity of Mycielski graph of G is more than that of G or not. Here we give bounds for the boxicity of Mycielski graphs in terms of the number of universal vertices of G and the edge clique cover number of the complement of G. Further observations determine the boxicity of the Mycielski graph M(G) if G has no universal vertices or odd universal vertices and box(G) is equal to the edge clique cover number of the complement of G. We also present relations between the Mycielski graph M(G) and its generalizations M3(G) and Mr(G) in the context of boxicity, which will encourage us to calculate the boxicity of M(G) and M3(G).

Properties, proved and conjectured, of Keller, Mycielski, and queen graphs

We prove several results about three families of graphs. For queen graphs, defined from the usual moves of a chess queen, we find the edge-chromatic number in almost all cases. In the unproved case, we have a conjecture supported by a vast amount of computation, which involved the development of a new edge-coloring algorithm. The conjecture is that the edge-chromatic number is the maximum degree, except when simple arithmetic forces the edge-chromatic number to be one greater than the maximum degree. For Mycielski graphs, we strengthen an old result that the graphs are Hamiltonian by showing that they are Hamilton-connected (except M 3 , which is a cycle). For Keller graphs G d , we establish, in all cases, the exact value of the chromatic number, the edge-chromatic number, and the independence number; and we get the clique covering number in all cases except 5 ≤ d ≤ 7 . We also investigate Hamiltonian decompositions of Keller graphs, obtaining them up to G 6 .

On ordinary and weighted Harary indices of Mycielski graphs

The Mycielski’s construction is a well-known construction on graphs which transforms a [Formula: see text]-chromatic triangle-free graph [Formula: see text] into a [Formula: see text]-chromatic triangle-free graph [Formula: see text] which is called the Mycielski graph of [Formula: see text]. In this paper, we compute the Harary index, additively weighted Harary index, and multiplicatively weighted Harary index of the Mycielski graph of any simple connected graph with girth greater than [Formula: see text].

Tree‐Chromatic Number Is Not Equal to Path‐Chromatic Number*

AbstractFor a graph G and a tree‐decomposition of G, the chromatic number of is the maximum of , taken over all bags . The tree‐chromatic number of G is the minimum chromatic number of all tree‐decompositions of G. The path‐chromatic number of G is defined analogously. In this article, we introduce an operation that always increases the path‐chromatic number of a graph. As an easy corollary of our construction, we obtain an infinite family of graphs whose path‐chromatic number and tree‐chromatic number are different. This settles a question of Seymour (J Combin Theory Ser B 116 (2016), 229–237). Our results also imply that the path‐chromatic numbers of the Mycielski graphs are unbounded.

Generalised Mycielski graphs, signature systems, and bounds on chromatic numbers

We prove that the coindex of the box complex B(H) of a graph H can be measured by the generalised Mycielski graphs which admit a homomorphism to H. As a consequence, we exhibit for every graph H a system of linear equations, solvable in polynomial time, with the following properties: if the system has no solutions, then coind(B(H))+2≤3; if the system has solutions, then χ(H)≥4. We generalise the method to other bounds on chromatic numbers using linear algebra.

Bondage Numbers of Mycielski Graphs

The bondage number \(b(G)\) of a nonempty graph \(G\) is the smallest number of edges, removal of which from \(G\) results in a graph with domination number greater than that of \(G\). In this paper, we study the bondage number of Mycielski graphs. First, we present some properties of dominating sets and give sharp upper bounds for bondage numbers of Mycielski graphs. Second, we obtain a characterization of Mycielski graphs of trees bondage numbers of which are 1. Finally, we calculate the exact values for bondage numbers of Mycielski graphs underlying graphs of which are complete graphs, paths, cycles, complete \(t\)-partite graphs, etc.

A note on total graphs

Erratum: Identication and corrections of the existing mistakes in the paper On the total graph of Mycielski graphs, central graphs and their covering numbers, Discuss. Math. Graph Theory 33 (2013) 361{371.

Colouring quadrangulations of projective spaces

A graph embedded in a surface with all faces of size 4 is known as a quadrangulation. We extend the definition of quadrangulation to higher dimensions, and prove that any graph G which embeds as a quadrangulation in the real projective space Pn has chromatic number n+2 or higher, unless G is bipartite. For n=2 this was proved by Youngs (1996) [20]. The family of quadrangulations of projective spaces includes all complete graphs, all Mycielski graphs, and certain graphs homomorphic to Schrijver graphs. As a corollary, we obtain a new proof of the Lovász–Kneser theorem.