scholarly journals Triangles in a complete chromatic graph

1985 ◽  
Vol 39 (1) ◽  
pp. 86-93 ◽  
Author(s):  
A. W. Goodman
Keyword(s):  
Complete Graph ◽  
Difficult Problem ◽  
Minimum Number

AbstractSuppose that in a complete graph on N points, each edge is given arbitrarily either the color red or the color blue, but the total number of blue edges is fixed at T. We find the minimum number of monochromatic triangles in the graph as a function of N and T. The maximum number of monochromatic triangles presents a more difficult problem. Here we propose a reasonable conjecture supported by examples.

scholarly journals An Improvement of the Lower Bound on the Minimum Number of ≤k-Edges

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 525
Author(s):  
Javier Rodrigo ◽  
Susana Merchán ◽  
Danilo Magistrali ◽  
Mariló López
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
General Position ◽  
Crossing Number ◽  
Minimum Number

In this paper, we improve the lower bound on the minimum number of  ≤k-edges in sets of n points in general position in the plane when k is close to n2. As a consequence, we improve the current best lower bound of the rectilinear crossing number of the complete graph Kn for some values of n.

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.

Component Edge Connectivity of Hypercubes

2018 ◽  
Vol 29 (06) ◽  
pp. 995-1001 ◽  
Author(s):  
Shuli Zhao ◽  
Weihua Yang ◽  
Shurong Zhang ◽  
Liqiong Xu
Keyword(s):  
Fault Tolerance ◽  
Complete Graph ◽  
Interconnection Networks ◽  
Interconnection Network ◽  
Optimal Solutions ◽  
Edge Connectivity ◽  
Important Measure ◽  
Minimum Number ◽  
Text Component

Fault tolerance is an important issue in interconnection networks, and the traditional edge connectivity is an important measure to evaluate the robustness of an interconnection network. The component edge connectivity is a generalization of the traditional edge connectivity. The [Formula: see text]-component edge connectivity [Formula: see text] of a non-complete graph [Formula: see text] is the minimum number of edges whose deletion results in a graph with at least [Formula: see text] components. Let [Formula: see text] be an integer and [Formula: see text] be the decomposition of [Formula: see text] such that [Formula: see text] and [Formula: see text] for [Formula: see text]. In this note, we determine the [Formula: see text]-component edge connectivity of the hypercube [Formula: see text], [Formula: see text] for [Formula: see text]. Moreover, we classify the corresponding optimal solutions.

A (5,5)-Colouring of Kn with Few Colours

2018 ◽  
Vol 27 (6) ◽  
pp. 892-912
Author(s):  
ALEX CAMERON ◽  
EMILY HEATH
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Minimum Number ◽  
Edge Colouring

For fixed integers p and q, let f(n,p,q) denote the minimum number of colours needed to colour all of the edges of the complete graph Kn such that no clique of p vertices spans fewer than q distinct colours. Any edge-colouring with this property is known as a (p,q)-colouring. We construct an explicit (5,5)-colouring that shows that f(n,5,5) ≤ n1/3 + o(1) as n → ∞. This improves upon the best known probabilistic upper bound of O(n1/2) given by Erdős and Gyárfás, and comes close to matching the best known lower bound Ω(n1/3).

scholarly journals An Explicit Edge-Coloring of $K_n$ with Six Colors on Every $K_5$

10.37236/7852 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Alex Cameron
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Edge Coloring ◽  
Minimum Number ◽  
Positive Integers

Let $p$ and $q$ be positive integers such that $1 \leq q \leq {p \choose 2}$. A $(p,q)$-coloring of the complete graph on $n$ vertices $K_n$ is an edge coloring for which every $p$-clique contains edges of at least $q$ distinct colors. We denote the minimum number of colors needed for such a $(p,q)$-coloring of $K_n$ by $f(n,p,q)$. This is known as the Erdös-Gyárfás function. In this paper we give an explicit $(5,6)$-coloring with $n^{1/2+o(1)}$ colors. This improves the best known upper bound of $f(n,5,6)=O\left(n^{3/5}\right)$ given by Erdös and Gyárfás, and comes close to matching the order of the best known lower bound, $f(n,5,6) = \Omega\left(n^{1/2}\right)$.

scholarly journals Ramsey Numbers of Ordered Graphs

10.37236/7816 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Martin Balko ◽  
Josef Cibulka ◽  
Karel Král ◽  
Jan Kynčl
Keyword(s):  
Chromatic Number ◽  
Complete Graph ◽  
Positive Answer ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Minimum Number ◽  
Ordered Graphs ◽  
Constant Interval ◽  
Monochromatic Copy ◽  
Special Classes

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of $\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by Károlyi, Pach, Tóth, and Valtr.

scholarly journals Minimum number of below average triangles in a weighted complete graph

2006 ◽  
Vol 3 (3) ◽  
pp. 206-219 ◽  
Author(s):  
Gareth Bendall ◽  
François Margot
Keyword(s):  
Complete Graph ◽  
Minimum Number

scholarly journals Removing Symmetry in Circulant Graphs and Point-Block Incidence Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 166
Author(s):  
Josephine Brooks ◽  
Alvaro Carbonero ◽  
Joseph Vargas ◽  
Rigoberto Flórez ◽  
Brendan Rooney ◽  
...  
Keyword(s):  
Graph Theory ◽  
Design Theory ◽  
Difficult Problem ◽  
Combinatorial Design ◽  
Surprising Result ◽  
Cyclic Structure ◽  
Circulant Graphs ◽  
Minimum Number ◽  
Automorphism Of A Graph

An automorphism of a graph is a mapping of the vertices onto themselves such that connections between respective edges are preserved. A vertex v in a graph G is fixed if it is mapped to itself under every automorphism of G. The fixing number of a graph G is the minimum number of vertices, when fixed, fixes all of the vertices in G. The determination of fixing numbers is important as it can be useful in determining the group of automorphisms of a graph-a famous and difficult problem. Fixing numbers were introduced and initially studied by Gibbons and Laison, Erwin and Harary and Boutin. In this paper, we investigate fixing numbers for graphs with an underlying cyclic structure, which provides an inherent presence of symmetry. We first determine fixing numbers for circulant graphs, showing in many cases the fixing number is 2. However, we also show that circulant graphs with twins, which are pairs of vertices with the same neighbourhoods, have considerably higher fixing numbers. This is the first paper that investigates fixing numbers of point-block incidence graphs, which lie at the intersection of graph theory and combinatorial design theory. We also present a surprising result-identifying infinite families of graphs in which fixing any vertex fixes every vertex, thus removing all symmetries from the graph.

scholarly journals Forcing a sparse minor

2015 ◽  
Vol 25 (2) ◽  
pp. 300-322 ◽  
Author(s):  
BRUCE REED ◽  
DAVID R. WOOD
Keyword(s):  
Complete Graph ◽  
Absolute Constant ◽  
Average Degree ◽  
Hadwiger's Conjecture ◽  
Minimum Number ◽  
A Minor ◽  
Hadwiger’S Conjecture

This paper addresses the following question for a given graphH: What is the minimum numberf(H) such that every graph with average degree at leastf(H) containsHas a minor? Due to connections with Hadwiger's conjecture, this question has been studied in depth whenHis a complete graph. Kostochka and Thomason independently proved that$f(K_t)=ct\sqrt{\ln t}$. More generally, Myers and Thomason determinedf(H) whenHhas a super-linear number of edges. We focus on the case whenHhas a linear number of edges. Our main result, which complements the result of Myers and Thomason, states that ifHhastvertices and average degreedat least some absolute constant, then$f(H)\leq 3.895\sqrt{\ln d}\,t$. Furthermore, motivated by the case whenHhas small average degree, we prove that ifHhastvertices andqedges, thenf(H) ⩽t+ 6.291q(where the coefficient of 1 in thetterm is best possible).

The Restricted Edge-Connectivity of Kronecker Product Graphs

2019 ◽  
Vol 29 (03) ◽  
pp. 1950012
Author(s):  
Tianlong Ma ◽  
Jinling Wang ◽  
Mingzu Zhang
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Kronecker Product ◽  
Sufficient Condition ◽  
Connected Component ◽  
Edge Connectivity ◽  
Product Graphs ◽  
Minimum Number ◽  
Vertex Set ◽  
Product Of Graphs

The restricted edge-connectivity of a connected graph [Formula: see text], denoted by [Formula: see text], if exists, is the minimum number of edges whose deletion disconnects the graph such that each connected component has at least two vertices. The Kronecker product of graphs [Formula: see text] and [Formula: see text], denoted by [Formula: see text], is the graph with vertex set [Formula: see text], where two vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. In this paper, it is proved that [Formula: see text] for any graph [Formula: see text] and a complete graph [Formula: see text] with [Formula: see text] vertices, where [Formula: see text] is minimum edge-degree of [Formula: see text], and a sufficient condition such that [Formula: see text] is [Formula: see text]-optimal is acquired.

Sign in / Sign up

Export Citation Format

Share Document