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 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 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.

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 Unimodality of the independence polynomials of some composite graphs

Filomat ◽  
2017 ◽  
Vol 31 (3) ◽  
pp. 629-637 ◽  
Author(s):  
Bao-Xuan Zhu ◽  
Qinglin Lu
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Lexicographic Product ◽  
Complete Multipartite Graph ◽  
Real Zeros ◽  
Independence Polynomial ◽  
Independence Polynomials ◽  
Multipartite Graph ◽  
Complete Multipartite

Let I(G;x) denote the independence polynomial of a graph G. In this paper we study the unimodality properties of I(G; x) for some composite graphs G. Given two graphs G1 and G2, let G1[G2] denote the lexicographic product of G1 and G2. Assume I(G1; x) = P i_0 aixi and I(G2; x) = P i_0 bixi, where I(G2; x) is log-concave. Then we prove (i) if I(G1; x) is logconcave and (a2i ??ai??1ai+1)b21 _ aiai??1b2 for all 1 _ i _ _(G1), then I(G1[G2]; x) is log-concave; (ii) if ai??1 _ b1ai for 1 _ i _ _(G1), then I(G1[G2]; x) is unimodal. In particular, if ai is increasing in i, then I(G1[G2]; x) is unimodal. We also give two su_cient conditions when the independence polynomial of a complete multipartite graph is unimodal or log-concave. Finally, for every odd positive integer _ > 3, we find a connected graph G not a tree, such that _(G) = _, and I(G; x) is symmetric and has only real zeros. This answers a problem of Mandrescu and Miric?a.

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.

scholarly journals On-Line Choice Number of Complete Multipartite Graphs: an Algorithmic Approach

10.37236/3378 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Fei-Huang Chang ◽  
Hong-Bin Chen ◽  
Jun-Yi Guo ◽  
Yu-Pei Huang
Keyword(s):  
Independence Number ◽  
Algorithmic Approach ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
On Line ◽  
Choice Number ◽  
Complete Multipartite

This paper studies the on-line choice number on complete multipartite graphs with independence number $m$. We give a unified strategy for every prescribed $m$. Our main result leads to several interesting consequences comparable to known results. (1) If $k_1-\sum_{p=2}^m\left(\frac{p^2}{2}-\frac{3p}{2}+1\right)k_p\geq 0$, where $k_p$ denotes the number of parts of cardinality $p$, then $G$ is on-line chromatic-choosable. (2) If $|V(G)|\leq\frac{m^2-m+2}{m^2-3m+4}\chi(G)$, then $G$ is on-line chromatic-choosable. (3) The on-line choice number of regular complete multipartite graphs $K_{m\star k}$ is at most$\left(m+\frac{1}{2}-\sqrt{2m-2}\right)k$ for $m\geq 3$.

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