scholarly journals Sharply Transitive $1$-Factorizations of Complete Multipartite Graphs

10.37236/349 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Giuseppe Mazzuoccolo ◽  
Gloria Rinaldi
Keyword(s):  
Complete Multipartite Graph ◽  
Transitive Action ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Vertex Set ◽  
Even Order ◽  
Multipartite Graph ◽  
A Finite Group ◽  
Complete Multipartite ◽  
Sharply Transitive

Given a finite group $G$ of even order, which graphs $\Gamma$ have a $1$-factorization admitting $G$ as automorphism group with a sharply transitive action on the vertex-set? Starting from this question, we prove some general results and develop an exhaustive analysis when $\Gamma$ is a complete multipartite graph and $G$ is cyclic.

scholarly journals On groups admitting no integral Cayley graphs besides complete multipartite graphs

2013 ◽  
Vol 7 (1) ◽  
pp. 119-128 ◽  
Author(s):  
Alireza Abdollahi ◽  
Mojtaba Jazaeri
Keyword(s):  
Cayley Graph ◽  
Identity Element ◽  
Simple Groups ◽  
Complete Multipartite Graph ◽  
Arbitrary Graph ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Vertex Set ◽  
Multipartite Graph ◽  
Complete Multipartite

Let G be a non-trivial finite group, S ? G \ {e} be a set such that if a 2 S, then a-1 ? S and e be the identity element of G. Suppose that Cay(G, S) is the Cayley graph with the vertex set G such that two vertices a and b are adjacent whenever a-1 ? S. An arbitrary graph is called integral whenever all eigenvalues of the adjacency matrix are integers. We say that a group G is Cayley integral simple whenever every connected integral Cayley graph on G is isomorphic to a complete multipartite graph. In this paper we prove that if G is a non-simple group, then G is Cayley integral simple if and only if G ? Zp2 for some prime number p or G ? Z2 x Z2. Moreover, we show that there exist finite non-abelian simple groups which are not Cayley integral simple.

scholarly journals A shorter proof of the distance energy of complete multipartite graphs

2017 ◽  
Vol 5 (1) ◽  
pp. 61-63 ◽  
Author(s):  
Wasin So
Keyword(s):  
Linear Algebra ◽  
Symmetric Matrix ◽  
Complete Multipartite Graph ◽  
Real Symmetric Matrix ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Multipartite Graph ◽  
Complete Multipartite

Abstract Caporossi, Chasser and Furtula in [Les Cahiers du GERAD (2009) G-2009-64] conjectured that the distance energy of a complete multipartite graph of order n with r ≥ 2 parts, each of size at least 2, is equal to 4(n − r). Stevanovic, Milosevic, Hic and Pokorny in [MATCH Commun. Math. Comput. Chem. 70 (2013), no. 1, 157-162.] proved the conjecture, and then Zhang in [Linear Algebra Appl. 450 (2014), 108-120.] gave another proof. We give a shorter proof of this conjecture using the interlacing inequalities of a positve semi-definite rank-1 perturbation to a real symmetric matrix.

scholarly journals On-Line List Colouring of Complete Multipartite Graphs

10.37236/2050 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Seog-Jin Kim ◽  
Young Soo Kwon ◽  
Daphne Der-Fen Liu ◽  
Xuding Zhu
Keyword(s):  
Positive Integer ◽  
Complete Multipartite Graph ◽  
Minimal Function ◽  
Line List ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
On Line ◽  
Multipartite Graph ◽  
Complete Multipartite

The Ohba Conjecture says that every graph $G$ with $|V(G)| \le 2 \chi(G)+1$ is chromatic choosable. This paper studies an on-line version of Ohba Conjecture. We prove that unlike the off-line case, for $k \ge 3$, the complete multipartite graph $K_{2\star (k-1), 3}$ is not on-line chromatic-choosable. Based on this result, the on-line version of Ohba Conjecture is modified as follows: Every graph $G$ with $|V(G)| \le 2 \chi(G)$ is on-line chromatic choosable. We present an explicit strategy to show  that for any positive integer $k$, the graph $K_{2\star k}$ is on-line chromatic-choosable.  We then present a minimal function $g$ for which the graph $K_{2 \star (k-1), 3}$ is on-line $g$-choosable.

scholarly journals The Erdős–Rothschild problem on edge-colourings with forbidden monochromatic cliques

2017 ◽  
Vol 163 (2) ◽  
pp. 341-356 ◽  
Author(s):  
OLEG PIKHURKO ◽  
KATHERINE STADEN ◽  
ZELEALEM B. YILMA
Keyword(s):  
Stability Theorem ◽  
Complete Multipartite Graph ◽  
Natural Numbers ◽  
Asymptotic Structure ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Multipartite Graph ◽  
Edge Colourings ◽  
Complete Multipartite

AbstractLet k := (k1,. . .,ks) be a sequence of natural numbers. For a graph G, let F(G;k) denote the number of colourings of the edges of G with colours 1,. . .,s such that, for every c ∈ {1,. . .,s}, the edges of colour c contain no clique of order kc. Write F(n;k) to denote the maximum of F(G;k) over all graphs G on n vertices. This problem was first considered by Erdős and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases.We prove that, for every k and n, there is a complete multipartite graph G on n vertices with F(G;k) = F(n;k). Also, for every k we construct a finite optimisation problem whose maximum is equal to the limit of log2F(n;k)/${n\choose 2}$ as n tends to infinity. Our final result is a stability theorem for complete multipartite graphs G, describing the asymptotic structure of such G with F(G;k) = F(n;k) · 2o(n2) in terms of solutions to the optimisation problem.

Codes from multipartite graphs and minimal permutation decoding sets

2015 ◽  
Vol 07 (04) ◽  
pp. 1550060
Author(s):  
P. Seneviratne
Keyword(s):  
Automorphism Group ◽  
Error Correcting Code ◽  
Binary Codes ◽  
Hard Problem ◽  
Complete Multipartite Graph ◽  
Decoding Algorithm ◽  
Permutation Decoding ◽  
Multipartite Graphs ◽  
Multipartite Graph ◽  
Complete Multipartite

Permutation decoding method developed by MacWilliams and described in [Permutation decoding of systematic codes, Bell Syst. Tech. J. 43 (1964) 485–505] is a decoding technique that uses a subset of the automorphism group of the code called a PD-set. The complexity of the permutation decoding algorithm depends on the size of the PD-set and finding a minimal PD-set for an error correcting code is a hard problem. In this paper we examine binary codes from the complete-multipartite graph [Formula: see text] and find PD-sets for all values of [Formula: see text] and [Formula: see text]. Further we show that these PD-sets are minimal when [Formula: see text] is odd and [Formula: see text].

THE EXISTENCE OR NONEXISTENCE OF NON-COMMUTING GRAPHS WITH PARTICULAR PROPERTIES

2013 ◽  
Vol 13 (01) ◽  
pp. 1350064 ◽  
Author(s):  
M. AKBARI ◽  
A. R. MOGHADDAMFAR
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Strongly Regular Graph ◽  
Complete Multipartite Graph ◽  
Commuting Graph ◽  
Central Elements ◽  
Vertex Set ◽  
Strongly Regular ◽  
Multipartite Graph ◽  
Complete Multipartite

We consider the non-commuting graph ∇(G) of a non-abelian finite group G; its vertex set is G\Z(G), the set of non-central elements of G, and two distinct vertices x and y are joined by an edge if [x, y] ≠ 1. We determine the structure of any finite non-abelian group G (up to isomorphism) for which ∇(G) is a complete multipartite graph (see Propositions 3 and 4). It is also shown that a non-commuting graph is a strongly regular graph if and only if it is a complete multipartite graph. Finally, it is proved that there is no non-abelian group whose non-commuting graph is self-complementary and n-cube.

scholarly journals Cyclic and Dihedral 1-Factorizations of Multipartite Graphs

10.37236/666 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Mathieu Bogaerts ◽  
Giuseppe Mazzuoccolo
Keyword(s):  
Automorphism Group ◽  
Dihedral Group ◽  
Existence Problem ◽  
Complete Multipartite Graph ◽  
Complete Answer ◽  
Multipartite Graphs ◽  
Graph Mapping ◽  
Multipartite Graph ◽  
Complete Multipartite

An automorphism group $G$ of a $1$-factorization of the complete multipartite graph $K_{m\times n}$ consists of permutations of the vertices of the graph mapping factors to factors. In this paper, we give a complete answer to the existence problem of a $1$-factorization of $K_{m\times n}$ admitting a cyclic or dihedral group acting sharply transitively on the vertices of the graph.

On the local irregularity vertex coloring of volcano, broom, parachute, double broom and complete multipartite graphs

2021 ◽  
Author(s):  
Arika Indah Kristiana ◽  
Nafidatun Nikmah ◽  
Dafik ◽  
Ridho Alfarisi ◽  
M. Ali Hasan ◽  
...  
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Vertex Coloring ◽  
Minimum Cardinality ◽  
Label Function ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Vertex Set ◽  
Complete Multipartite ◽  
Local Irregularity

Let [Formula: see text] be a simple, finite, undirected, and connected graph with vertex set [Formula: see text] and edge set [Formula: see text]. A bijection [Formula: see text] is label function [Formula: see text] if [Formula: see text] and for any two adjacent vertices [Formula: see text] and [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] is set ofvertices adjacent to [Formula: see text]. [Formula: see text] is called local irregularity vertex coloring. The minimum cardinality of local irregularity vertex coloring of [Formula: see text] is called chromatic number local irregular denoted by [Formula: see text]. In this paper, we verify the exact values of volcano, broom, parachute, double broom and complete multipartite graphs.

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