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.

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].

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 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-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 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 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.

scholarly journals The Size, Multipartite Ramsey Numbers for nK2 Versus Path–Path and Cycle

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 764
Author(s):  
Yaser Rowshan ◽  
Mostafa Gholami ◽  
Stanford Shateyi
Keyword(s):  
Positive Integer ◽  
Ramsey Number ◽  
Complete Multipartite Graph ◽  
Ramsey Numbers ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Monochromatic Copy

For given graphs G1,G2,…,Gn and any integer j, the size of the multipartite Ramsey number mj(G1,G2,…,Gn) is the smallest positive integer t such that any n-coloring of the edges of Kj×t contains a monochromatic copy of Gi in color i for some i, 1≤i≤n, where Kj×t denotes the complete multipartite graph having j classes with t vertices per each class. In this paper, we computed the size of the multipartite Ramsey numbers mj(K1,2,P4,nK2) for any j,n≥2 and mj(nK2,C7), for any j≤4 and n≥2.

scholarly journals The Energy Change of the Complete Multipartite Graph

2020 ◽  
Vol 36 (36) ◽  
pp. 309-317
Author(s):  
Haiying Shan ◽  
Changxiang He ◽  
Zhensheng Yu
Keyword(s):  
Linear Algebra ◽  
Singular Values ◽  
New Method ◽  
Energy Change ◽  
Natural Question ◽  
Complete Multipartite Graph ◽  
Multipartite Graph ◽  
Energy Of Graph ◽  
Complete Multipartite ◽  
Edge Addition

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. Akbari et al. [S. Akbari, E. Ghorbani, and M. Oboudi. Edge addition, singular values, and energy of graphs and matrices. {\em Linear Algebra Appl.}, 430:2192--2199, 2009.] proved that for a complete multipartite graph $K_{t_1 ,\ldots,t_k}$, if $t_i\geq 2 \ (i=1,\ldots,k)$, then deleting any edge will increase the energy. A natural question is how the energy changes when $\min\{t_1 ,\ldots,t_k\}=1$. In this paper, a new method to study the energy of graph is explored. As an application of this new method, the above natural question is answered and it is completely determined how the energy of a complete multipartite graph changes when one edge is removed.

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