Union Of Cycles Research Articles

Complete bipartite graphs without small rainbow stars

The k-edge-colored bipartite Gallai–Ramsey number bgrk(G:H) is defined as the minimum integer n such that n2≥k and for every N≥n, every edge-coloring (using all k colors) of complete bipartite graph KN,N contains a rainbow copy of G or a monochromatic copy of H. In this paper, we first study the structural theorem on the complete bipartite graph Kn,n with no rainbow copy of K1,3. Next, we utilize the results to prove the exact values of bgrk(P4:H), bgrk(P5:H), bgrk(K1,3:H), where H is a various union of cycles and paths and stars.

FIBONACCI AND SUPER FIBONACCI GRACEFUL LABELLINGS OF SOME TYPES OF GRAPHS

We consider the basic theoretical information regarding the Fibonacci graceful graphs. An injective function is said a Fibonacci graceful labelling of a graph of a size , if it induces a bijective function on the set of edges , where by the rule , for any adjacent vertices A graph that allows such labelling is called Fibonacci graceful. In this paper, we introduce the concept of super Fibonacci graceful labelling, narrowing the set of vertex labels, i.e. Four types of problems to be studied are selected. In the problem of the first type, the following question is raised: is there a graph that allows a certain kind of labelling, and under what conditions does this take place? The problem of the second type is the problem of construction: it is necessary, for a given system of requirements for the graph, to construct (at least one) its labelling that would satisfy this system. The following two types of problems relate to enumeration problems: for a given graph, determine the number of different Fibonacci and / or super Fibonacci graceful labellings; build all the different labellings of a given kind. As a result of solving these problems, functions were found that generate Fibonacci and super Fibonacci graceful labellings for graphs of cyclic structure; necessary and sufficient conditions for the existence of Fibonacci graceful labelling for disjunctive union of cycles, super Fibonacci graceful labelling for cycles, Eulerian graphs are obtained; the number of non-equivalent labellings of the cycle is determined; conditions for the existence of a super Fibonacci graceful labelling of a one-point connection of arbitrary connected super Fibonacci graceful graphs … …, are presented

Asymptotic density of graphs excluding disconnected minors

For a graph H, letc∞(H)=limn→∞⁡max⁡|E(G)|n, where the maximum is taken over all graphs G on n vertices not containing H as a minor. Thus c∞(H) is the asymptotic maximum density of graphs not containing H as a minor. Employing a structural lemma due to Eppstein, we prove new upper bounds on c∞(H) for disconnected graphs H. In particular, we determine c∞(H) whenever H is a union of cycles. Finally, we investigate the behaviour of c∞(sKr) for fixed r, where sKr denotes the union of s disjoint copies of the complete graph on r vertices. Improving on a result of Thomason, we show thatc∞(sKr)=s(r−1)−1fors=ω(log⁡rlog⁡log⁡r), andc∞(sKr)>s(r−1)−1fors=o(log⁡rlog⁡log⁡r).

Edge magic total labeling of lexicographic product C4(2r+1) o ~K2 cycle with chords, unions of paths, and unions of cycles and paths

<p>An <em>edge magic total (EMT) labeling</em> of a graph <span class="math"><em>G</em> = (<em>V</em>, <em>E</em>)</span> is a bijection from the set of vertices and edges to a set of numbers defined by <span class="math"><em>λ</em> : <em>V</em> ∪ <em>E</em> → {1, 2, ..., ∣<em>V</em>∣ + ∣<em>E</em>∣}</span> with the property that for every <span class="math"><em>x</em><em>y</em> ∈ <em>E</em></span>, the weight of <span class="math"><em>x</em><em>y</em></span> equals to a constant <span class="math"><em>k</em></span>, that is, <span class="math"><em>λ</em>(<em>x</em>) + <em>λ</em>(<em>y</em>) + <em>λ</em>(<em>x</em><em>y</em>) = <em>k</em></span> for some integer <span class="math"><em>k</em></span>. In this paper given the construction of an EMT labeling for certain lexicographic product <span class="math">$C_{4(2r+1)}\circ \overline{K_2}$</span>, cycle with chords <span class="math"><em></em><sup>[<em>c</em>]<em>t</em></sup><em>C</em><sub><em>n</em></sub></span>, unions of paths <span class="math"><em>m</em><em>P</em><sub><em>n</em></sub></span>, and unions of cycles and paths <span class="math"> <em>m</em>(<em>C</em><sub><em>n</em><sub>1</sub>(2<em>r</em> + 1)</sub> ∪ (2<em>r</em> + 1)<em>P</em><sub><em>n</em><sub>2</sub></sub>)</span>.</p>

Integer-antimagic spectra of disjoint unions of cycles

Integer-antimagic spectra of disjoint unions of cycles

Prime Valuation of Union of Cycles and Cycle Related Graphs

Prime Valuation of Union of Cycles and Cycle Related Graphs

The extremal function for disconnected minors

For a graph H let c(H) denote the supremum of |E(G)|/|V(G)| taken over all non-null graphs G not containing H as a minor. We show thatc(H)≤|V(H)|+comp(H)2−1, when H is a union of cycles. This verifies a conjecture of Reed and Wood, and another conjecture of Harvey and Wood.We derive the above result from a theorem which allows us to find two vertex-disjoint subgraphs with prescribed densities in a sufficiently dense graph, which might be of independent interest.

Packing chromatic number, $$\mathbf (1, 1, 2, 2) $$ ( 1 , 1 , 2 , 2 ) -colorings, and characterizing the Petersen graph

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets \(\Pi _1,\ldots ,\Pi _k\), where \(\Pi _i\), \(i\in [k]\), is an i-packing. The following conjecture is posed and studied: if G is a subcubic graph, then \(\chi _{\rho }(S(G))\le 5\), where S(G) is the subdivision of G. The conjecture is proved for all generalized prisms of cycles. To get this result it is proved that if G is a generalized prism of a cycle, then G is (1, 1, 2, 2)-colorable if and only if G is not the Petersen graph. The validity of the conjecture is further proved for graphs that can be obtained from generalized prisms in such a way that one of the two n-cycles in the edge set of a generalized prism is replaced by a union of cycles among which at most one is a 5-cycle. The packing chromatic number of graphs obtained by subdividing each of its edges a fixed number of times is also considered.

On the dynamics of endomorphisms of finite groups

Aiming at a better understanding of finite groups as finite dynamical systems, we show that by a version of Fitting’s Lemma for groups, each state space of an endomorphism of a finite group is a graph tensor product of a finite directed 1-tree whose cycle is a loop with a disjoint union of cycles, generalizing results of Hernández-Toledo on linear finite dynamical systems, and we fully characterize the possible forms of state spaces of nilpotent endomorphisms via their “ramification behavior”. Finally, as an application, we will count the isomorphism types of state spaces of endomorphisms of finite cyclic groups in general, extending results of Hernández-Toledo on primary cyclic groups of odd order.

Some Edge Pair Sum Graphs

An injective map f : E(G)→{± 1,±2, … ,±q} is said to be an edge pair sum labeling of a graph G(p, q) if the induced vertex function f* : V(G)→Z − {0} defined by f* (v)=∑e∈Evf(e) is one – one, where Ev denotes the set of edges in G that are incident with a vertex v and f*(V(G)) is either of the form or according as p is even or odd. A graph which admits edge pair sum labeling is called an edge pair sum graph. In this paper we prove that the one point union of cycles, the perfect binary tree, shadow graph, total graph and Pn2 admit edge pair sum labeling.

Some Graceful Graphs

In this paper, we have investigate some general classes of connected graceful graphs. We prove that cycles $C_{n}, C_{m}$ joined by an arbitrary path $P_{k}$, where $m,n \equiv 0$ (mod $4$), a cycle $C_{n}$, ($n \equiv 2$ (mod $4$)) with twin chords and a path union of cycles $C_{n}$, ($n \equiv 0$ (mod $4$)) are graceful graphs.

Triangular matrix coalgebras and applications

We formally introduce and study generalized comatrix coalgebras and upper triangular comatrix coalgebras, which are not only a dualization but also an extension of classical generalized matrix algebras. We use these to answer several open questions on Noetherian and Artinian type notions in the theory of coalgebras, and to give complete connections between these. We also solve completely the so called finite splitting problem for coalgebras: we show that is a coalgebra such that the rational part of every left finitely generated -module splits off if and only if is an upper triangular matrix coalgebra, for a serial coalgebra whose Ext-quiver is a finite union of cycles, a finite dimensional coalgebra and a finite dimensional bicomodule .

The f-Chromatic Index of a Graph Whose f-Core Has Maximum Degree 2

Abstract.Let G be a graph. The minimum number of colors needed to color the edges of G is called the chromatic index of G and is denoted by χ'(G). It is well known that , for any graph G, where Δ(G) denotes the maximum degree of G. A graph G is said to be class 1 if x'(G) = Δ(G) and class 2 if χ'(G) = Δ(G)+1. Also, GΔ is the induced subgraph on all vertices of degree Δ(G). Let f : V(G) → ℕ be a function. An f-coloring of a graph G is a coloring of the edges of E(G) such that each color appears at each vertex v ∊ V(G) at most f (v) times. The minimum number of colors needed to f-color G is called the f-chromatic index of G and is denoted by χ'f (G). It was shown that for every graph , where . A graph G is said to be f -class 1 , and f -class 2, otherwise. Also, GΔf is the induced subgraph of G on . Hilton and Zhao showed that if G has maximum degree two and G is class 2, then G is critical, GΔ is a disjoint union of cycles and δ(G) = Δ(G)–1, where δ(G) denotes the minimum degree of G, respectively. In this paper, we generalize this theorem to f -coloring of graphs. Also, we determine the f -chromatic index of a connected graph G with |GΔf| ≤ 4.

A CONJUGATION-FREE GEOMETRIC PRESENTATION OF FUNDAMENTAL GROUPS OF ARRANGEMENTS II: EXPANSION AND SOME PROPERTIES

A conjugation-free geometric presentation of a fundamental group is a presentation with the natural topological generators x1,…,xn and the cyclic relations: [Formula: see text] with no conjugations on the generators. We have already proved in [13] that if the graph of the arrangement is a disjoint union of cycles, then its fundamental group has a conjugation-free geometric presentation. In this paper, we extend this property to arrangements whose graphs are a disjoint union of cycle-tree graphs. Moreover, we study some properties of this type of presentations for a fundamental group of a line arrangement's complement. We show that these presentations satisfy a completeness property in the sense of Dehornoy, if the corresponding graph of the arrangement is triangle-free. The completeness property is a powerful property which leads to many nice properties concerning the presentation (such as the left-cancellativity of the associated monoid and yields some simple criterion for the solvability of the word problem in the group).

THE EXISTENCE OF A MAJORITY RULE MAXIMAL SET IN ARBITRARY n-DIMENSIONAL SPATIAL MODELS

We examine the effect of indifference on the existence of a majority rule maximal set. In our setting, it is shown in all but a limited number of cases that the maximal set is empty in an n-dimensional spatial model if and only if the Pareto set contains a union of cycles. The elements that constitute the exception are completely characterized.

The Distribution of the Size of the Union of Cycles for Two Types of Random Permutations

We discuss some problems and permutation statistics involving two different types of random permutations. Under the usual model of random permutations, we prove that the shifted coverage of the elements of {1, 2, …, k} of a random permutation over {1, 2, …, n}; that is, the size of the union of the cycles containing these elements, excluding these elements themselves, follows a negative hypergeometric distribution. This fact gives a probabilistic model for the coverage via the canonical cycle representation. For a different random model, we determine some random permutation statistics regarding the problem of the lost boarding pass and its variations.

On ?-Fold Equipartite Oberwolfach Problem with Uniform Table Sizes

We consider the following generalization of the Oberwolfach problem: "At a gathering there are n delegations each having m people. Is it possible to arrange a seating of mn people present at s round tables T1, T2,...,Ts (where each Ti can accommodate ti ≥ 3 people and ∑ti = mn )f ork different meals so that each person has every other person not in the same delegation for a neighbor exactly λ times?" For λ= 1, Liu has obtained the complete solution to the problem when all tables accommodate the same number t of people. In this paper, we give the complete solution to the problem for λ ≥ 2 when all tables have uniform sizes t. Throughout the paper, we use Cn for a cycle on n vertices, Pn for a path on n vertices, Kn for the complete graph on n vertices, and K(m : n) for the complete n-partite graph with m vertices in each partite set (also called complete equipartite graph), namely, K(m : n )= K(m1, m2,...,mn) with m1 = m2 = ··· = mn = m. Also, for a graph G ,w e use λG to represent the multi-graph obtained from G by replacing each edge of G with λ copies of it. A fa ctor Fof a graph G is a subgraph for which V(F )= V(G) .A nr- fa ctorof G is a factor that is regular of degree r. Clearly, a 2-factor is a disjoint union of cycles. An r- f actorization of a graph G is a partition of the edge set E(G) into r- fa ctors. Thus, a graph G having a 2-factorization must be regular of even degree. An {H1, H2,...,Hk}- factorization of a graph G is a partition of the edge set E(G) into factors such that each component of any factor is isomorphic to Hi for some 1 ≤ i ≤ k. In particular, an H- f actorization of G is a partition of E(G) into factors such that each factor is a disjoint union of H's. Consequently, a Ct -factorization of G is a 2-factorization with each 2-factor being a disjoint union of Ct 's.

On Graceful Directed Graphs

A digraph D with e arcs is numbered by assigning a distinct integer value $\theta ( v )$ from $\{ 0,1, \cdots ,e \}$ to each node v. The node values, in turn, induce a value $\theta ( u,v )$ on each arc $( u,v )$ where $\theta ( u,v ) = \theta ( v ) - \theta ( u )\bmod ( e + 1 )$. If the arc values are all distinct and nonzero, then the numbering is graceful. A digraph is graceful if it has a graceful numbering. Graceful digraphs are related in a variety of ways to other areas of mathematics. It is shown here that the graceful numberings of certain classes of digraphs are characterized by the existence of particular algebraic structures, including cyclic difference sets, sequenceable groups, $( K,1 )$ complete mappings, and $( K,1)$ near-complete mappings. Some digraph models of cyclic groups are examined here, and cyclic neofields are shown to generate families of graceful numberings for the models. Properties and examples of this new class of graph numberings are presented here. For instance, families of graceful digraphs include certain orientations of cycles, paths, and the unions of cycles and paths, as well as certain complete digraphs, wheels, windmills, and umbrellas. Techniques are developed by whichdigraphs can be embedded as subgraphs of infinite families of graceful digraphs. A variety of fundamental questions are posed and some conjectures advanced.

Modular lattices in which each element is a union of cycles

Modular lattices in which each element is a union of cycles