scholarly journals On the Buratti-Horak-Rosa Conjecture about Hamiltonian Paths in Complete Graphs

10.37236/3879 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Anita Pasotti ◽  
Marco Antonio Pellegrini
Keyword(s):  
Positive Integer ◽  
Complete Graph ◽  
Complete Solution ◽  
Complete Graphs ◽  
Hamiltonian Paths

In this paper we investigate a problem proposed by Marco Buratti, Peter Horak and Alex Rosa (denoted by BHR-problem) concerning Hamiltonian paths in the complete graph with prescribed edge-lengths. In particular we solve BHR$(\{1^a, 2^b, t^c\})$ for any even integer $t \geq 4$,  provided that $a+b \geq t-1$. Furthermore, for $t=4, 6, 8$ we present a complete solution of BHR$(\{ 1^a,2^b,t^c \})$ for any positive integer $a,b,c$.

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.

A study on distance in graph complement

2020 ◽  
Vol 12 (03) ◽  
pp. 2050045
Author(s):  
A. Chellaram Malaravan ◽  
A. Wilson Baskar
Keyword(s):  
Positive Integer ◽  
Complete Graph ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

The aim of this paper is to determine radius and diameter of graph complements. We provide a necessary and sufficient condition for the complement of a graph to be connected, and determine the components of graph complement. Finally, we completely characterize the class of graphs [Formula: see text] for which the subgraph induced by central (respectively peripheral) vertices of its complement in [Formula: see text] is isomorphic to a complete graph [Formula: see text], for some positive integer [Formula: see text].

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.

scholarly journals On the Unicity Conjecture for Generalized Markoff Equation

2018 ◽  
Vol 3 ◽  
pp. 137 ◽  
Author(s):  
Youb Raj Gaire
Keyword(s):  
Positive Integer ◽  
Complete Solution ◽  
Partial Solution ◽  
Famous Conjecture ◽  
Markoff Equation ◽  
College Of Engineering ◽  
Positive Integers

<p>The Markoff equation x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = 3xyz is introduced by A.A. Markoff in 1879. A famous conjecture on the Markoff equation, made by Frobinus in 1913, states that any Markoff triples (x, y, z) with x ≤ y ≤ z is uniquely determined by its largest number z. The complete solution of this equation is still open however the partial solution is given by Barager (1996), Button (2001), Zhang (2007), Srinivasan (2009), Chen and Chen (2013). In 1957, Mordell developed a generalization to the Markoff equation of the form x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = Axyz + B where, A and B are positive integers. In 2015, Donald McGinn take a particular form of above equation with A = 1 and B = A and gave a partial solution to the unicity conjecture to this equation. In this paper, the partial solution to the unicity conjecture to the equation of the form x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> =3xyz + A where A is positive integer with A ≤ 4(x<sup>2</sup> –1) is given. </p><p><strong>Journal of Advanced College of Engineering and Management,</strong> Vol. 3, 2017, Page : 137-145</p>

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.

XVII Algebraical researches, containing a disquisition on Newton’s rule for the discovery of imaginary roots, and an allied rule applicable to a particular class of equations, together with a complete invariantive determination of the character of the roots of the general equation of the fifth degree, &c

1864 ◽  
Vol 154 ◽  
pp. 579-666 ◽  
Keyword(s):  
Positive Integer ◽  
Empirical Evidence ◽  
General Equation ◽  
Complete Solution ◽  
Present Form ◽  
The Third ◽  
Free Running ◽  
Inferior Limit ◽  
Imaginary Roots

(1) This memoir in its present form is of the nature of a trilogy; it is divided into three parts, of which each has its action complete within itself, but the same general cycle of ideas pervades all three, and weaves them into a sort of complex unity. In the first is established the validity of Newton’s rule for finding an inferior limit to the number of imaginary roots of algebraical equations as far as the fifth degree inclusive. In the second is obtained a rule for assigning a like limit applicable to equations of the form Σ( ax + b ) m =0, m being any positive integer, and the coefficients a , b real. In the third are determined the absolute invariantive criteria for fixing unequivocally the character of the roots of an equation of the fifth degree, that is to say, for ascertaining the exact number of real and imaginary roots which it contains. This last part has been added since the original paper was presented to the Society. It has grown out of a foot-note appended to the second, itself an independent offshoot from the first part, hut may be studied in a great measure independently of what precedes, and constitutes, in the author’s opinion, by far the most valuable portion of the memoir, containing as it does a complete solution of one of the most interesting and fruitful algebraical questions which has ever yet engaged the attention of mathematicians (1). I propose in a subse­quent addition to the memoir to resume and extend some of the investigations which incidentally arise in this part. The foot-notes are numbered and lettered for facility of reference, and will be found in many instances of equal value with the matter in the text, to which they serve as a kind of free running accompaniment and commentary. 2) In the ‘Arithmetica Universalis,’ in the first chapter on equations, Newton has given a rule for discovering an inferior limit to the number of imaginary roots in an equation of any degree, without proof or indication of the method by which he arrived at it, or the evidence upon which it rests(²). Maclaurin, in vol. xxxiv. p. 104, and vol. xxxvi. p. 59 of the Philosophical Transactions, Campbell (³) in vol. xxxviii. p. 515 of the same, and other authors of reputation have sought in vain for a demonstration of this marvellous and mysterious rule ( 4 ). Unwilling to rest my belief in it on mere empirical evidence, I have investigated and obtained a demonstration of its truth as far as the fifth degree inclusive, which, although presenting only a small instalment of the desired result, I am induced to offer for insertion in the Transactions in the hope of exciting renewed attention to a subject so intimately bound up with the fundamental principles of algebra.

scholarly journals TILING DIRECTED GRAPHS WITH TOURNAMENTS

2018 ◽  
Vol 6 ◽  
Author(s):  
ANDRZEJ CZYGRINOW ◽  
LOUIS DEBIASIO ◽  
THEODORE MOLLA ◽  
ANDREW TREGLOWN
Keyword(s):  
Positive Integer ◽  
Directed Graph ◽  
Complete Graph ◽  
Directed Graphs ◽  
Type Result ◽  
Vertex Disjoint

The Hajnal–Szemerédi theorem states that for any positive integer $r$ and any multiple $n$ of $r$, if $G$ is a graph on $n$ vertices and $\unicode[STIX]{x1D6FF}(G)\geqslant (1-1/r)n$, then $G$ can be partitioned into $n/r$ vertex-disjoint copies of the complete graph on $r$ vertices. We prove a very general analogue of this result for directed graphs: for any positive integer $r$ with $r\neq 3$ and any sufficiently large multiple $n$ of $r$, if $G$ is a directed graph on $n$ vertices and every vertex is incident to at least $2(1-1/r)n-1$ directed edges, then $G$ can be partitioned into $n/r$ vertex-disjoint subgraphs of size $r$ each of which contain every tournament on $r$ vertices (the case $r=3$ is different and was handled previously). In fact, this result is a consequence of a tiling result for standard multigraphs (that is multigraphs where there are at most two edges between any pair of vertices). A related Turán-type result is also proven.

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 Note on Odd Cycle-Complete Graph Ramsey Numbers

10.37236/1662 ◽  
2001 ◽  
Vol 9 (1) ◽  
Author(s):  
Benny Sudakov
Keyword(s):  
Positive Integer ◽  
Complete Graph ◽  
Upper Bound ◽  
Short Note ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Odd Cycle

The Ramsey number $r(C_l, K_n)$ is the smallest positive integer $m$ such that every graph of order $m$ contains either cycle of length $l$ or a set of $n$ independent vertices. In this short note we slightly improve the best known upper bound on $r(C_l, K_n)$ for odd $l$.

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