scholarly journals Bent Hamilton Cycles in $d$-Dimensional Grid Graphs

10.37236/1694 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
F. Ruskey ◽  
Joe Sawada
Keyword(s):  
Hamilton Cycle ◽  
Graph Product ◽  
Grid Graph ◽  
Hamilton Cycles ◽  
Hamilton Path ◽  
Successive Pair ◽  
Grid Graphs

A bent Hamilton cycle in a grid graph is one in which each edge in a successive pair of edges lies in a different dimension. We show that the $d$-dimensional grid graph has a bent Hamilton cycle if some dimension is even and $d \geq 3$, and does not have a bent Hamilton cycle if all dimensions are odd. In the latter case, we determine the conditions for when a bent Hamilton path exists. For the $d$-dimensional toroidal grid graph (i.e., the graph product of $d$ cycles), we show that there exists a bent Hamilton cycle when all dimensions are odd and $d \geq 3$. We also show that if $d=2$, then there exists a bent Hamilton cycle if and only if both dimensions are even.

scholarly journals Counting Hamilton cycles in Dirac hypergraphs

2020 ◽  
pp. 1-23
Author(s):  
Stefan Glock ◽  
Stephen Gould ◽  
Felix Joos ◽  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Similar Estimate ◽  
Hamilton Cycles

Abstract A tight Hamilton cycle in a k-uniform hypergraph (k-graph) G is a cyclic ordering of the vertices of G such that every set of k consecutive vertices in the ordering forms an edge. Rödl, Ruciński and Szemerédi proved that for $k\ge 3$ , every k-graph on n vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such k-graphs is ${\exp(n\ln n-\Theta(n))}$ . As a corollary, we obtain a similar estimate on the number of Hamilton ${\ell}$ -cycles in such k-graphs for all ${\ell\in\{0,\ldots,k-1\}}$ , which makes progress on a question of Ferber, Krivelevich and Sudakov.

scholarly journals An approximate version of Jackson’s conjecture

2020 ◽  
Vol 29 (6) ◽  
pp. 886-899
Author(s):  
Anita Liebenau ◽  
Yanitsa Pehova
Keyword(s):  
Bipartite Graph ◽  
Small Proportion ◽  
Complete Bipartite Graph ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Edge Disjoint ◽  
Bipartite Digraph ◽  
Bipartite Tournament ◽  
Approximate Version

AbstractA diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and out-degree. In 1981 Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact its edge set can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: for every ε > 0 there exists n0 such that every diregular bipartite tournament on 2n ≥ n0 vertices contains a collection of (1/2–ε)n cycles of length at least (2–ε)n. Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for every c > 1/2 and ε > 0 there exists n0 such that every cn-regular bipartite digraph on 2n ≥ n0 vertices contains (1−ε)cn edge-disjoint Hamilton cycles.

Visual/Graphic Aids for the Technical Report

1979 ◽  
Vol 9 (3) ◽  
pp. 287-291 ◽  
Author(s):  
Robert Cury
Keyword(s):  
Flow Chart ◽  
White Space ◽  
Grid Graph ◽  
Grid Graphs ◽  
Process Maps ◽  
Technical Report ◽  
Technical Writer ◽  
Flow Charts ◽  
Visual Description ◽  
Visual Graphic

Authors of technical papers have many visual/graphic aids available to them. The most common are: grid graphs, tables, bar charts, flow charts, maps, pie diagrams, and drawings and sketches. Grid graphs are used to show relationships. Tables allow the reader to make comparisons of data. The bar chart is another form of the grid graph and is used for the same purpose. A flow chart gives the reader a visual description of a process. Maps show the location of specific features. Pie diagrams show the proportional breakdown of a topic. Pictures and sketches show the reader exactly what is being talked about in the report. Visual/graphic aids allow the technical writer to condense and present his information in an aesthetically pleasing manner; in addition, these aids serve as psychological white space.

Nonexistence of a pair of arc disjoint directed Hamilton cycles on line digraphs of 2-diregular digraphs

2015 ◽  
Vol 07 (03) ◽  
pp. 1550034
Author(s):  
T. Govindan ◽  
A. Muthusamy
Keyword(s):  
Line Graph ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
On Line

Bermond conjectured that if G is Hamilton cycle decomposable, then L(G), the line graph of G is Hamilton cycle decomposable. In this paper, we prove that, for any k > 5, there exists a directed Hamilton cycle decomposable 2-diregular digraph D of order 2k such that L(D) is not directed Hamilton cycle decomposable.

scholarly journals Multi-Coloured Hamilton Cycles in Random Edge-Coloured Graphs

2002 ◽  
Vol 11 (2) ◽  
pp. 129-133 ◽  
Author(s):  
COLIN COOPER ◽  
ALAN FRIEZE
Keyword(s):  
Hamilton Cycle ◽  
Hamilton Cycles

We define a space of random edge-coloured graphs [Gscr ]n,m,κ which correspond naturally to edge κ-colourings of Gn,m. We show that there exist constants K0, K1 [les ] 21 such that, provided m [ges ] K0n log n and κ [ges ] K1n, then a random edge-coloured graph contains a multi-coloured Hamilton cycle with probability tending to 1 as the number of vertices n tends to infinity.

ALTERNATING HAMILTON CYCLES WITH MINIMUM NUMBER OF CROSSINGS IN THE PLANE

2000 ◽  
Vol 10 (01) ◽  
pp. 73-78 ◽  
Author(s):  
ATSUSHI KANEKO ◽  
M. KANO ◽  
KIYOSHI YOSHIMOTO
Keyword(s):  
Upper Bound ◽  
Time Algorithm ◽  
Hamilton Cycle ◽  
Crossing Number ◽  
Hamilton Cycles ◽  
Line Segments ◽  
Straight Line ◽  
Minimum Number ◽  
Disjoint Sets

Let X and Y be two disjoint sets of points in the plane such that |X|=|Y| and no three points of X ∪ Y are on the same line. Then we can draw an alternating Hamilton cycle on X∪Y in the plane which passes through alternately points of X and those of Y, whose edges are straight-line segments, and which contains at most |X|-1 crossings. Our proof gives an O(n2 log n) time algorithm for finding such an alternating Hamilton cycle, where n =|X|. Moreover we show that the above upper bound |X|-1 on crossing number is best possible for some configurations.

scholarly journals Edge Disjoint Hamilton Cycles in Knödel Graphs

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Palanivel Subramania Nadar Paulraja ◽  
S Sampath Kumar
Keyword(s):  
Maximum Degree ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Cycle Decomposition ◽  
Edge Disjoint ◽  
International Audience ◽  
Appl Math

International audience The vertices of the Knödel graph $W_{\Delta, n}$ on $n \geq 2$ vertices, $n$ even, and of maximum degree $\Delta, 1 \leq \Delta \leq \lfloor log_2(n) \rfloor$, are the pairs $(i,j)$ with $i=1,2$ and $0 \leq j \leq \frac{n}{2} -1$. For $0 \leq j \leq \frac{n}{2} -1$, there is an edge between vertex $(1,j)$ and every vertex $(2,j + 2^k - 1 (mod \frac{n}{2}))$, for $k=0,1,2, \ldots , \Delta -1$. Existence of a Hamilton cycle decomposition of $W_{k, 2k}, k \geq 6$ is not yet known, see Discrete Appl. Math. 137 (2004) 173-195. In this paper, it is shown that the $k$-regular Knödel graph $W_{k,2k}, k \geq 6$ has $ \lfloor \frac{k}{2} \rfloor - 1$ edge disjoint Hamilton cycles.

scholarly journals Loose Hamilton Cycles in Random 3-Uniform Hypergraphs

10.37236/477 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Alan Frieze
Keyword(s):  
Absolute Constant ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Uniform Hypergraphs ◽  
Random Hypergraph

In the random hypergraph $H=H_{n,p;3}$ each possible triple appears independently with probability $p$. A loose Hamilton cycle can be described as a sequence of edges $\{x_i,y_i,x_{i+1}\}$ for $i=1,2,\ldots,n/2$ where $x_1,x_2,\ldots,x_{n/2},y_1,y_2,\ldots,y_{n/2}$ are all distinct. We prove that there exists an absolute constant $K>0$ such that if $p\geq {K\log n\over n^2}$ then $$\lim_{\textstyle{n\to \infty\atop 4|n}}\Pr(H_{n,p;3}\ contains\ a\ loose\ Hamilton\ cycle)=1.$$

scholarly journals Spanning Trees of Bounded Degree

10.37236/1577 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Andrzej Czygrinow ◽  
Genghua Fan ◽  
Glenn Hurlbert ◽  
H. A. Kierstead ◽  
William T. Trotter
Keyword(s):  
Spanning Tree ◽  
Maximum Degree ◽  
Additional Assumption ◽  
Spanning Trees ◽  
Independent Set ◽  
Hamilton Cycle ◽  
Bounded Degree ◽  
Hamilton Path ◽  
Classic Theorem

Dirac's classic theorem asserts that if ${\bf G}$ is a graph on $n$ vertices, and $\delta({\bf G})\ge n/2$, then ${\bf G}$ has a hamilton cycle. As is well known, the proof also shows that if $\deg(x)+\deg(y)\ge(n-1)$, for every pair $x$, $y$ of independent vertices in ${\bf G}$, then ${\bf G}$ has a hamilton path. More generally, S. Win has shown that if $k\ge 2$, ${\bf G}$ is connected and $\sum_{x\in I}\deg(x)\ge n-1$ whenever $I$ is a $k$-element independent set, then ${\bf G}$ has a spanning tree ${\bf T}$ with $\Delta({\bf T})\le k$. Here we are interested in the structure of spanning trees under the additional assumption that ${\bf G}$ does not have a spanning tree with maximum degree less than $k$. We show that apart from a single exceptional class of graphs, if $\sum_{x\in I}\deg(x)\ge n-1$ for every $k$-element independent set, then ${\bf G}$ has a spanning caterpillar ${\bf T}$ with maximum degree $k$. Furthermore, given a maximum path $P$ in ${\bf G}$, we may require that $P$ is the spine of ${\bf T}$ and that the set of all vertices whose degree in ${\bf T}$ is $3$ or larger is independent in ${\bf T}$.

scholarly journals Computing the Domination Number of Grid Graphs

10.37236/628 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Samu Alanko ◽  
Simon Crevals ◽  
Anton Isopoussu ◽  
Patric Östergård ◽  
Ville Pettersson
Keyword(s):  
Dynamic Programming ◽  
Dominating Set ◽  
Domination Number ◽  
Dynamic Programming Algorithm ◽  
Programming Algorithm ◽  
Dominating Sets ◽  
Grid Graph ◽  
Grid Graphs ◽  
Square Grid ◽  
Minimum Dominating Set

Let $\gamma_{m,n}$ denote the size of a minimum dominating set in the $m \times n$ grid graph. For the square grid graph, exact values for $\gamma_{n,n}$ have earlier been published for $n \leq 19$. By using a dynamic programming algorithm, the values of $\gamma_{m,n}$ for $m,n \leq 29$ are here obtained. Minimum dominating sets for square grid graphs up to size $29 \times 29$ are depicted.

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