The Coarseness of the Complete Graph

1968 ◽  
Vol 20 ◽  
pp. 888-894 ◽  
Author(s):  
Richard K. Guy ◽  
Lowell W. Beineke
Keyword(s):  
Complete Graph ◽  
Integer Part ◽  
Edge Disjoint ◽  
Image Position ◽  
Planar Subgraphs

The coarseness, c(G), of a graph G is the maximum number of edge-disjoint, non-planar subgraphs of G. We consider only the complete graph, Kp, on p vertices here. For p = 3r, Erdös conjectured that the coarseness was , but it has been shown (1) that1where square brackets denote integer part.

The Coarseness of the Complete Bipartite Graph

1969 ◽  
Vol 21 ◽  
pp. 1086-1096 ◽  
Author(s):  
Lowell W. Beineke ◽  
Richard K. Guy
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
The Other ◽  
Integer Part ◽  
Edge Disjoint ◽  
Image Position

The coarseness, c(G), of a graph G is the maximum number of edge-disjoint, non-planar graphs whose union is G. The coarseness of the complete graph has been investigated elsewhere (1; 2). We consider the coarseness of the complete bipartite, or 2-coloured, graph, Km,n, consisting of sets of mand nvertices, each member of one set being joined by an edge to each member of the other. No members of one set are joined to each other.Our results are summarized in the following theorem, where square brackets denote “integer part”.THEOREM. If m= 3p + d, 0 ≦ d≦ 2, and n = 3q + e, 0 ≦ e ≦ 2, then for d = 0 or 1 and e = 0 or 1,1

scholarly journals On the Length of a Random Minimum Spanning Tree

2015 ◽  
Vol 25 (1) ◽  
pp. 89-107 ◽  
Author(s):  
COLIN COOPER ◽  
ALAN FRIEZE ◽  
NATE INCE ◽  
SVANTE JANSON ◽  
JOEL SPENCER
Keyword(s):  
Complete Graph ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Expected Value ◽  
Edge Weight ◽  
Formula Group ◽  
Image Position

We study the expected value of the lengthLnof the minimum spanning tree of the complete graphKnwhen each edgeeis given an independent uniform [0, 1] edge weight. We sharpen the result of Frieze [6] that limn→∞$\mathbb{E}$(Ln) = ζ(3) and show that$$ \mathbb{E}(L_n)=\zeta(3)+\frac{c_1}{n}+\frac{c_2+o(1)}{n^{4/3}}, $$wherec1,c2are explicitly defined constants.

k-Degenerate Graphs

1970 ◽  
Vol 22 (5) ◽  
pp. 1082-1096 ◽  
Author(s):  
Don R. Lick ◽  
Arthur T. White
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
Natural Numbers ◽  
Image Position ◽  
Disconnected Graphs ◽  
Totally Disconnected

Graphs possessing a certain property are often characterized in terms of a type of configuration or subgraph which they cannot possess. For example, a graph is totally disconnected (or, has chromatic number one) if and only if it contains no lines; a graph is a forest (or, has point-arboricity one) if and only if it contains no cycles. Chartrand, Geller, and Hedetniemi [2] defined a graph to have property Pn if it contains no subgraph homeomorphic from the complete graph Kn+1 or the complete bipartite graphFor the first four natural numbers n, the graphs with property Pn are exactly the totally disconnected graphs, forests, outerplanar and planar graphs, respectively. This unification suggested the extension of many results known to hold for one of the above four classes of graphs to one or more of the remaining classes.

Bounds on the Coarseness of the n-Cube

1979 ◽  
Vol 22 (2) ◽  
pp. 171-175 ◽  
Author(s):  
Jehuda Hartman
Keyword(s):  
Edge Disjoint ◽  
Dimensional Cube ◽  
Image Position

AbstractThe coarseness, c(G), of a graph G is the maximum number of edge disjoint nonplanar subgraphs contained in G For the n-dimensional cube Qn we obtain the inequalities

scholarly journals A result for semi-regular continued fractions

1969 ◽  
Vol 10 (1-2) ◽  
pp. 145-154 ◽  
Author(s):  
P. E. Blanksby
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Two Dimensional ◽  
Integer Part ◽  
Image Position

If Φ is a real number with |Φ| ≧ 1, then a semiregular continuet fraction development of Φ is denoted by where the ai are integers such that |ai| ≧ 2. The expansions arise geo-. metrically by considering the sequence of divided cells of two-dimensional grids (see [1]), and are described by the following algorithm: for all n ≧ 0, taking Φ = Φ.0 Hence where in this case the square brackets are used to signify the integer-part function. It follows that each irrational Φ has uncountably many such expansions, none of which has a constantly equal to 2 (or -2) for large n.

scholarly journals Time-discretised Galerkin approximations of parabolic stochastic PDE's

1996 ◽  
Vol 54 (1) ◽  
pp. 79-85 ◽  
Author(s):  
W. Grecksch ◽  
P.E. Kloeden
Keyword(s):  
Stochastic Partial Differential Equation ◽  
Time Interval ◽  
Integer Part ◽  
Initial Value ◽  
Time Step ◽  
Finite Dimensional ◽  
Galerkin Approximations ◽  
Discretisation Error ◽  
Image Position ◽  
Taylor Scheme

The global discretisation error is estimated for strong time discretisations of finite dimensional Ito stochastic differential equations (SDEs) which are Galerkin approximations of a class of parabolic stochastic partial differential equation (SPDE) with a strongly monotone linear operator with eigenvalues λ1 ≤ λ2 ≤ … in its drift term. If an order γ strong Taylor scheme with time-step δ is applied to the N dimensional Ito-Galerkin SDE, the discretisation error is bounded above bywhere [x] is the integer part of the real number x and the constant K depends on the initial value, bounds on the other coefficients in the SPDE and the length of the time interval under consideration.

scholarly journals Clumsy Packings of Graphs

10.37236/7942 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Maria Axenovich ◽  
Anika Kaufmann ◽  
Raphael Yuster
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Minimum Size ◽  
Maximum Cardinality ◽  
Natural Parameter ◽  
Free Graph ◽  
Minimum Cardinality ◽  
Empty Graph ◽  
Edge Disjoint ◽  
Hexagonal Grids

Let $G$ and $H$ be graphs. We say that $P$ is an $H$-packing of $G$ if $P$ is a set of edge-disjoint copies of $H$ in $G$. An $H$-packing $P$ is maximal if there is no other $H$-packing of $G$ that properly contains P. Packings of maximum cardinality have been studied intensively, with several recent breakthrough results. Here, we consider minimum cardinality maximal packings. An $H$-packing $P$ is called clumsy if it is maximal of minimum size. Let $\mathrm{cl}(G,H)$ be the size of a clumsy $H$-packing of $G$. We provide nontrivial bounds for $\mathrm{cl}(G,H)$, and in many cases asymptotically determine $\mathrm{cl}(G,H)$ for some generic classes of graphs G such as $K_n$ (the complete graph), $Q_n$ (the cube graph), as well as square, triangular, and hexagonal grids. We asymptotically determine $\mathrm{cl}(K_n,H)$ for every fixed non-empty graph $H$. In particular, we prove that  $$\mathrm{cl}(K_n, H) = \frac{\binom{n}{2}- \mathrm{ex}(n,H)}{|E(H)|}+o(\mathrm{ex}(n,H)),$$where $ex(n,H)$ is the extremal number of $H$. A related natural parameter is $\mathrm{cov}(G,H)$, that is the smallest number of copies of $H$ in $G$ (not necessarily edge-disjoint) whose removal from $G$ results in an $H$-free graph. While clearly $\mathrm{cov}(G,H) \leqslant\mathrm{cl}(G,H)$, all of our lower bounds for $\mathrm{cl}(G,H)$ apply to $\mathrm{cov}(G,H)$ as well.

scholarly journals Decompositions of Complete Graphs into Bipartite 2-Regular Subgraphs

10.37236/4634 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Darryn Bryant ◽  
Andrea Burgess ◽  
Peter Danziger
Keyword(s):  
Regular Graph ◽  
Complete Graph ◽  
Perfect Matching ◽  
Necessary Conditions ◽  
Complete Graphs ◽  
Edge Disjoint

It is shown that if $G$ is any bipartite 2-regular graph of order at most $n/2$ or at least $n-2$, then the obvious necessary conditions are sufficient for the existence of a decomposition of the complete graph of order $n$ into a perfect matching and edge-disjoint copies of $G$.

ANOTHER ARITHMETIC OF THE EVEN AND THE ODD

2018 ◽  
Vol 11 (3) ◽  
pp. 604-608
Author(s):  
CELIA SCHACHT
Keyword(s):  
Unary Operation ◽  
Axiom System ◽  
Integer Part ◽  
Image Position

AbstractThis article presents an axiom system for an arithmetic of the even and the odd, one that is stronger than those discussed in Pambuccian (2016) and Menn & Pambuccian (2016). It consists of universal sentences in a language extending the usual one with 0, 1, +, ·, <, – with the integer part of the half function $[{ \cdot \over 2}]$, and two unary operation symbols.

scholarly journals On Edge-Disjoint Spanning Trees in a Randomly Weighted Complete Graph

2017 ◽  
Vol 27 (2) ◽  
pp. 228-244 ◽  
Author(s):  
ALAN FRIEZE ◽  
TONY JOHANSSON
Keyword(s):  
Complete Graph ◽  
Spanning Trees ◽  
Total Weight ◽  
Edge Disjoint

Assume that the edges of the complete graphKnare given independent uniform [0, 1] weights. We consider the expected minimum total weightμkofk⩽ 2 edge-disjoint spanning trees. Whenkis large we show thatμk≈k2. Most of the paper is concerned with the casek= 2. We show thatm2tends to an explicitly defined constant and thatμ2≈ 4.1704288. . . .

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