Topological covers of complete graphs

1998 ◽  
Vol 123 (3) ◽  
pp. 549-559 ◽  
Author(s):  
A. GARDINER ◽  
CHERYL E. PRAEGER
Keyword(s):  
Complete Graph ◽  
Additional Assumption ◽  
Complete Graphs ◽  
Symmetric Graph ◽  
Quotient Graph ◽  
Vertex Set ◽  
Acts 2

Let Γ be a connected G-symmetric graph of valency r, whose vertex set V admits a non-trivial G-partition [Bscr ], with blocks B∈[Bscr ] of size v and with k[les ]v independent edges joining each pair of adjacent blocks. In a previous paper we introduced a framework for analysing such graphs Γ in terms of (a) the natural quotient graph Γ[Bscr ] of valency b=vr/k, and (b) the 1-design [Dscr ](B) induced on each block. Here we examine the case where k=v and Γ[Bscr ]=Kb+1 is a complete graph. The 1-design [Dscr ](B) is then degenerate, so gives no information: we therefore make the additional assumption that the stabilizer G(B) of the block B acts 2-transitively on B. We prove that there is then a unique exceptional graph for which [mid ]B[mid ]=v>b+1.

scholarly journals A Family of Symmetric Graphs with Complete Quotients

10.37236/5701 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Teng Fang ◽  
Xin Gui Fang ◽  
Binzhou Xia ◽  
Sanming Zhou
Keyword(s):  
Complete Graph ◽  
Incidence Structure ◽  
Finite Graph ◽  
Complete Classification ◽  
Complete Graphs ◽  
Quotient Graph ◽  
Point Set ◽  
Natural Incidence ◽  
Ordered Pairs

A finite graph $\Gamma$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on $V(\Gamma)$ and transitively on the set of ordered pairs of adjacent vertices of $\Gamma$. If $V(\Gamma)$ admits a nontrivial $G$-invariant partition ${\cal B}$ such that for blocks $B, C \in {\cal B}$ adjacent in the quotient graph $\Gamma_{{\cal B}}$ relative to ${\cal B}$, exactly one vertex of $B$ has no neighbour in $C$, then we say that $\Gamma$ is an almost multicover of $\Gamma_{{\cal B}}$. In this case there arises a natural incidence structure ${\cal D}(\Gamma, {\cal B})$ with point set ${\cal B}$. If in addition $\Gamma_{{\cal B}}$ is a complete graph, then ${\cal D}(\Gamma, {\cal B})$ is a $(G, 2)$-point-transitive and $G$-block-transitive $2$-$(|{\cal B}|, m+1, \lambda)$ design for some $m \geq 1$, and moreover either $\lambda=1$ or $\lambda=m+1$. In this paper we classify such graphs in the case when $\lambda = m+1$; this together with earlier classifications when $\lambda = 1$ gives a complete classification of almost multicovers of complete graphs.

scholarly journals Zumkeller Labeling of Complete Graphs

2019 ◽  
Vol 9 (1S3) ◽  
pp. 352-355
Keyword(s):  
Complete Graph ◽  
Vertex Function ◽  
Complete Graphs ◽  
Natural Numbers ◽  
Vertex Set ◽  
Fan Graph

Let G (V, E) be a graph with vertex set V and edge set E. The process of assigning natural numbers to the vertices of G such that the product of the numbers of adjacent vertices of G is a Zumkeller number on the edges of G is known as Zumkeller labeling of G. This can be achieved by defining an appropriate vertex function of G. In this article, we show the existence of this labeling to complete graph and fan graph.

Supermagic Complete Graphs

1967 ◽  
Vol 19 ◽  
pp. 427-438 ◽  
Author(s):  
B. M. Stewart
Keyword(s):  
Complete Graph ◽  
Complete Graphs ◽  
Vertex Set ◽  
Image Position

In our paper “Magic graphs” (1) we showed that every complete graph Kn with n ⩾ 5 is “magic,” i.e., if the vertex set is indicated {vi} and if eij is the edge joining vi and vj, i ≠ j , then there exists a function α(eij) such that the set {α(eij)} consists of distinct positive rational integers and the vertex sums1have a constant value σ(α) for k = 1, 2, … , n. We noted that K2 is magic and showed that K3 and K4 are not magic.

Minimal Decompositions of Complete Graphs into Subgraphs with Embeddability Properties

1969 ◽  
Vol 21 ◽  
pp. 992-1000 ◽  
Author(s):  
L. W. Beineke
Keyword(s):  
Projective Plane ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Graphs ◽  
Minimum Number ◽  
Double Torus

Although the problem of finding the minimum number of planar graphs into which the complete graph can be decomposed remains partially unsolved, the corresponding problem can be solved for certain other surfaces. For three, the torus, the double-torus, and the projective plane, a single proof will be given to provide the solutions. The same questions will also be answered for bicomplete graphs.

scholarly journals INTRINSICALLY LINKED GRAPHS WITH KNOTTED COMPONENTS

2012 ◽  
Vol 21 (07) ◽  
pp. 1250065 ◽  
Author(s):  
THOMAS FLEMING
Keyword(s):  
Complete Graph ◽  
Cartesian Product ◽  
Complete Graphs ◽  
Linking Number ◽  
A Minor

We construct a graph G such that any embedding of G into R3 contains a nonsplit link of two components, where at least one of the components is a nontrivial knot. Further, for any m < n we produce a graph H so that every embedding of H contains a nonsplit n component link, where at least m of the components are nontrivial knots. We then turn our attention to complete graphs and show that for any given n, every embedding of a large enough complete graph contains a 2-component link whose linking number is a nonzero multiple of n. Finally, we show that if a graph is a Cartesian product of the form G × K2, it is intrinsically linked if and only if G contains one of K5, K3,3 or K4,2 as a minor.

Finding a Longest Alternating Cycle in a 2-edge-coloured Complete Graph is in RP

1996 ◽  
Vol 5 (3) ◽  
pp. 297-306 ◽  
Author(s):  
Rachid Saad
Keyword(s):  
Complete Graph ◽  
Strong Evidence ◽  
General Setting ◽  
Maximum Length ◽  
Sufficient Condition ◽  
Complete Graphs ◽  
Random Polynomial ◽  
Exact Matching ◽  
Matching Problem ◽  
Bipartite Tournament

Jackson [10] gave a polynomial sufficient condition for a bipartite tournament to contain a cycle of a given length. The question arises as to whether deciding on the maximum length of a cycle in a bipartite tournament is polynomial. The problem was considered by Manoussakis [12] in the slightly more general setting of 2-edge coloured complete graphs: is it polynomial to find a longest alternating cycle in such coloured graphs? In this paper, strong evidence is given that such an algorithm exists. In fact, using a reduction to the well known exact matching problem, we prove that the problem is random polynomial.

Perfect 1-factorizations

2019 ◽  
Vol 69 (3) ◽  
pp. 479-496 ◽  
Author(s):  
Alexander Rosa
Keyword(s):  
Pairwise Disjoint ◽  
Hamiltonian Cycle ◽  
Perfect Matching ◽  
Complete Graphs ◽  
Regular Graphs ◽  
Cubic Graphs ◽  
Vertex Set ◽  
Early Survey

AbstractLetGbe a graph with vertex-setV=V(G) and edge-setE=E(G). A 1-factorofG(also calledperfect matching) is a factor ofGof degree 1, that is, a set of pairwise disjoint edges which partitionsV. A 1-factorizationofGis a partition of its edge-setEinto 1-factors. For a graphGto have a 1-factor, |V(G)| must be even, and for a graphGto admit a 1-factorization,Gmust be regular of degreer, 1 ≤r≤ |V| − 1.One can find in the literature at least two extensive surveys [69] and [89] and also a whole book [90] devoted to 1-factorizations of (mainly) complete graphs.A 1-factorization ofGis said to beperfectif the union of any two of its distinct 1-factors is a Hamiltonian cycle ofG. An early survey on perfect 1-factorizations (abbreviated as P1F) of complete graphs is [83]. In the book [90] a whole chapter (Chapter 16) is devoted to perfect 1-factorizations of complete graphs.It is the purpose of this article to present what is known to-date on P1Fs, not only of complete graphs but also of other regular graphs, primarily cubic graphs.

Characterizing 2-distance graphs

2019 ◽  
Vol 12 (01) ◽  
pp. 1950006 ◽  
Author(s):  
Ramuel P. Ching ◽  
I. J. L. Garces
Keyword(s):  
Complete Graph ◽  
Distance Graph ◽  
Simple Graph ◽  
Distance Graphs ◽  
Vertex Set

Let [Formula: see text] be a finite simple graph. The [Formula: see text]-distance graph [Formula: see text] of [Formula: see text] is the graph with the same vertex set as [Formula: see text] and two vertices are adjacent if and only if their distance in [Formula: see text] is exactly [Formula: see text]. A graph [Formula: see text] is a [Formula: see text]-distance graph if there exists a graph [Formula: see text] such that [Formula: see text]. In this paper, we give three characterizations of [Formula: see text]-distance graphs, and find all graphs [Formula: see text] such that [Formula: see text] or [Formula: see text], where [Formula: see text] is an integer, [Formula: see text] is the path of order [Formula: see text], and [Formula: see text] is the complete graph of order [Formula: see text].

Magic Valuations of Finite Graphs

1970 ◽  
Vol 13 (4) ◽  
pp. 451-461 ◽  
Author(s):  
Anton Kotzig ◽  
Alexander Rosa
Keyword(s):  
Undirected Graph ◽  
Complete Graphs ◽  
Finite Graphs ◽  
Vertex Set ◽  
Multiple Edges

The purpose of this paper is to investigate for graphs the existence of certain valuations which have some "magic" property. The question about the existence of such valuations arises from the investigation of another kind of valuations which are introduced in [1] and are related to cyclic decompositions of complete graphs into isomorphic subgraphs.Throughout this paper the word graph will mean a finite undirected graph without loops or multiple edges having at least one edge. By G(m, n) we denote a graph having m vertices and n edges, by V(G) and E(G) the vertex-set and the edge-set of G, respectively. Both vertices and edges are called the elements of the graph.

scholarly journals Perfect One-Factorization Conjecture

d'CARTESIAN ◽  
2015 ◽  
Vol 4 (1) ◽  
pp. 114
Author(s):  
Chriestie Montolalu
Keyword(s):  
Complete Graph ◽  
Complete Graphs

Perfect one-factorization of the complete graph K2n for all n greater and equal to 2 is conjectured. Nevertheless some families of complete graphs were found to have perfect one-factorization. This paper will show some of the perfect one-factorization results in some families of complete graph as well as some result in application. Keywords: complete graph, one-factorization

Sign in / Sign up

Export Citation Format

Share Document