scholarly journals One-Factorisations of Complete Graphs Constructed in Desarguesian Planes of Certain Odd Square Orders

10.37236/8378 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Nicola Pace ◽  
Angelo Sonnino
Keyword(s):  
Complete Graph ◽  
Affine Group ◽  
Geometric Construction ◽  
Complete Graphs ◽  
Desarguesian Plane ◽  
Desarguesian Planes

A geometric construction of one-factorisations of complete graphs $K_{q(q-1)}$ is provided for the case when either $q=2^d +1$ is a Fermat prime, or $q=9$. This construction uses the affine group $\mathrm{AGL}(1,q)$, points and ovals in the Desarguesian plane $\mathrm{PG}(2,q^{2})$ to produce one-factorisations of the complete graph $K_{q(q-1)}$.

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.

Veblen–Wedderburn hybrid planes

1969 ◽  
Vol 309 (1497) ◽  
pp. 157-170 ◽  
Keyword(s):  
Projective Plane ◽  
Galois Field ◽  
Real Field ◽  
Type I ◽  
Type Ii ◽  
Desarguesian Plane ◽  
The Real ◽  
Desarguesian Planes ◽  
Hughes Plane

The paper describes a method of redistributing the points of the collinear sets in a Desarguesian plane so as to produce a (hybrid) projective plane which is non-Desarguesian. The method is applied to the construction: (i) of a plane over a prescribed subfield of the real field, and (ii) of a plane (over a Galois field) which is proved to be identical with the Hughes plane. On the basis of this construction algebraic relations in the field can be interpreted as incidence relations in the hybrid plane. In order to verify that the planes of type (i) are not isomorphic with Desarguesian planes, some conditions are established which show that all planes of this type (as well as of type (ii)) contain Fano subplanes.

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

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.

Certain 3-decompositions of complete graphs, with an application to finite fields

1985 ◽  
Vol 99 (3-4) ◽  
pp. 277-281 ◽  
Author(s):  
Peter Rowlinson
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Complete Graph ◽  
Disjoint Union ◽  
Necessary Condition ◽  
Complete Graphs ◽  
Characteristic P ◽  
Strongly Regular

SynopsisA necessary condition is obtained for a complete graph to have a decomposition as the line-disjoint union of three isomorphic strongly regular subgraphs. The condition is used to determine the number of non-trivial solutions of the equation x3+y3 = z3 in a finite field of characteristic p ≡ 2 mod 3.

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 $Star^1$-convex functions on tropical linear spaces of complete graphs

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Laura Escobar
Keyword(s):  
Complete Graph ◽  
Piecewise Linear ◽  
Negative Answer ◽  
Linear Functions ◽  
One Dimension ◽  
Complete Graphs ◽  
Piecewise Linear Functions ◽  
Linear Spaces ◽  
International Audience ◽  
Tropical Linear Space

International audience Given a fan $\Delta$ and a cone $\sigma \in \Delta$ let $star^1(\sigma )$ be the set of cones that contain $\sigma$ and are one dimension bigger than $\sigma$ . In this paper we study two cones of piecewise linear functions defined on $\delta$ : the cone of functions which are convex on $star^1(σ\sigma)$ for all cones, and the cone of functions which are convex on $star^1(σ\sigma)$ for all cones of codimension 1. We give nice combinatorial descriptions for these two cones given two different fan structures on the tropical linear space of complete graphs. For the complete graph $K_5$, we prove that with the finer fan subdivision the two cones are not equal, but with the coarser subdivision they are the same. This gives a negative answer to a question of Gibney-Maclagan that for the finer subdivision the two cones are the same. Soit $\Delta$ un fan, pour $\sigma \in \Delta$ nous définissons $star^1(\sigma )$ comme l'ensemble de cônes qui contiennent $\sigma$ dont la dimension est un de plus que la dimension de $\sigma$ . Nous étudions deux cônes d'applications linéaires par morceaux définis sur $\Delta$ : le cône de fonctions convexes sur$star^1(\sigma )$, où $\sigma \in \Delta$ est un cône quelconque, et le cône de fonctions convexes sur $star^1(σ\sigma)$ où σ est un cône de codimension 1. étant donnés deux structures sur l'espace tropical linéaire de graphes complets, nous donnons de beaux descriptions combinatoires des cônes décrits en haut. Pour le graphe complet $K_5$, on démontre que avec la subdivision en fans plus fine, les deux cônes sont différentes, mais avec la subdivision plus gros ils sont cônes sont les mêmes. Ce résultant réponde négativement une question de Gibney-Maclagan.

scholarly journals Some Graphs Determined by their Signless Laplacian (Distance) Spectra

2020 ◽  
Vol 36 (36) ◽  
pp. 461-472
Author(s):  
Chandrashekar Adiga ◽  
Kinkar Das ◽  
B. R. Rakshith
Keyword(s):  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Discrete Math ◽  
Complete Graphs ◽  
Spectral Determination ◽  
Regular Graphs ◽  
Bipartite Subgraph ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian

In literature, there are some results known about spectral determination of graphs with many edges. In [M.~C\'{a}mara and W.H.~Haemers. Spectral characterizations of almost complete graphs. {\em Discrete Appl. Math.}, 176:19--23, 2014.], C\'amara and Haemers studied complete graph with some edges deleted for spectral determination. In fact, they found that if the deleted edges form a matching, a complete graph $K_m$ provided $m \le n-2$, or a complete bipartite graph, then it is determined by its adjacency spectrum. In this paper, the graph $K_{n}\backslash K_{l,m}$ $(n>l+m)$ which is obtained from the complete graph $K_{n}$ by removing all the edges of a complete bipartite subgraph $K_{l,m}$ is studied. It is shown that the graph $K_{n}\backslash K_{1,m}$ with $m\ge4$ is determined by its signless Laplacian spectrum, and it is proved that the graph $K_{n}\backslash K_{l,m}$ is determined by its distance spectrum. The signless Laplacian spectral determination of the multicone graph $K_{n-2\alpha}\vee \alpha K_{2}$ was studied by Bu and Zhou in [C.~Bu and J.~Zhou. Signless Laplacian spectral characterization of the cones over some regular graphs. {\em Linear Algebra Appl.}, 436:3634--3641, 2012.] and Xu and He in [L. Xu and C. He. On the signless Laplacian spectral determination of the join of regular graphs. {\em Discrete Math. Algorithm. Appl.}, 6:1450050, 2014.] only for $n-2\alpha=1 ~\text{or}~ 2$. Here, this problem is completely solved for all positive integer $n-2\alpha$. The proposed approach is entirely different from those given by Bu and Zhou, and Xu and He.

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