scholarly journals Fast Strategies in Waiter-Client Games

10.37236/9451 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Dennis Clemens ◽  
Pranshu Gupta ◽  
Fabian Hamann ◽  
Alexander Haupt ◽  
Mirjana Mikalački ◽  
...  
Keyword(s):  
Complete Graph ◽  
Spanning Trees ◽  
Perfect Matchings ◽  
Hamilton Cycles ◽  
The Family ◽  
Winning Set ◽  
Pancyclic Graphs

Waiter-Client games are played on some hypergraph $(X,\mathcal{F})$, where $\mathcal{F}$ denotes the family of winning sets. For some bias $b$, during each round of such a game Waiter offers to Client $b+1$ elements of $X$, of which Client claims one for himself while the rest go to Waiter. Proceeding like this Waiter wins the game if she forces Client to claim all the elements of any winning set from $\mathcal{F}$. In this paper we study fast strategies for several Waiter-Client games played on the edge set of the complete graph, i.e. $X=E(K_n)$, in which the winning sets are perfect matchings, Hamilton cycles, pancyclic graphs, fixed spanning trees or factors of a given graph.

scholarly journals Anti-Ramsey Numbers in Complete k-Partite Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Jili Ding ◽  
Hong Bian ◽  
Haizheng Yu
Keyword(s):  
Spanning Trees ◽  
Edge Coloring ◽  
Perfect Matchings ◽  
Ramsey Number ◽  
Hamilton Cycles ◽  
Ramsey Numbers ◽  
The Family

The anti-Ramsey number ARG,H is the maximum number of colors in an edge-coloring of G such that G contains no rainbow subgraphs isomorphic to H. In this paper, we discuss the anti-Ramsey numbers ARKp1,p2,…,pk,Tn, ARKp1,p2,…,pk,ℳ, and ARKp1,p2,…,pk,C of Kp1,p2,…,pk, where Tn,ℳ, and C denote the family of all spanning trees, the family of all perfect matchings, and the family of all Hamilton cycles in Kp1,p2,…,pk, respectively.

The Numbers of Spanning Trees, Hamilton Cycles and Perfect Matchings in a Random Graph

1994 ◽  
Vol 3 (1) ◽  
pp. 97-126 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Random Graph ◽  
Spanning Trees ◽  
Directed Graphs ◽  
Projection Methods ◽  
Bipartite Graphs ◽  
Perfect Matchings ◽  
Hamilton Cycles ◽  
Asymptotically Normal ◽  
Wide Range ◽  
Log Normal

The numbers of spanning trees, Hamilton cycles and perfect matchings in a random graph Gnm are shown to be asymptotically normal if m is neither too large nor too small. At the lowest limit m ≍ n3/2, these numbers are asymptotically log-normal. For Gnp, the numbers are asymptotically log-normal for a wide range of p, including p constant. The same results are obtained for random directed graphs and bipartite graphs. The results are proved using decomposition and projection methods.

scholarly journals On the number of spanning trees of K_n^m #x00B1 G graphs

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Stavros D. Nikolopoulos ◽  
Charis Papadopoulos
Keyword(s):  
Complete Graph ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Star Graph ◽  
Complete Multipartite Graph ◽  
The Family ◽  
International Audience ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Number Of Spanning Trees

International audience The K_n-complement of a graph G, denoted by K_n-G, is defined as the graph obtained from the complete graph K_n by removing a set of edges that span G; if G has n vertices, then K_n-G coincides with the complement øverlineG of the graph G. In this paper we extend the previous notion and derive determinant based formulas for the number of spanning trees of graphs of the form K_n^m #x00b1 G, where K_n^m is the complete multigraph on n vertices with exactly m edges joining every pair of vertices and G is a multigraph spanned by a set of edges of K_n^m; the graph K_n^m + G (resp. K_n^m - G) is obtained from K_n^m by adding (resp. removing) the edges of G. Moreover, we derive determinant based formulas for graphs that result from K_n^m by adding and removing edges of multigraphs spanned by sets of edges of the graph K_n^m. We also prove closed formulas for the number of spanning tree of graphs of the form K_n^m #x00b1 G, where G is (i) a complete multipartite graph, and (ii) a multi-star graph. Our results generalize previous results and extend the family of graphs admitting formulas for the number of their spanning trees.

scholarly journals On the Number of Perfect Matchings in Random Lifts

2010 ◽  
Vol 19 (5-6) ◽  
pp. 791-817 ◽  
Author(s):  
CATHERINE GREENHILL ◽  
SVANTE JANSON ◽  
ANDRZEJ RUCIŃSKI
Keyword(s):  
Asymptotic Formula ◽  
Complete Graph ◽  
Pairwise Disjoint ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Lattice Points ◽  
Laplace’S Method ◽  
Second Moment ◽  
Multiple Dimensions ◽  
Laplace's Method

Let G be a fixed connected multigraph with no loops. A random n-lift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the n-sets corresponding to the endpoints of the edge. Let XG be the number of perfect matchings in a random lift of G. We study the distribution of XG in the limit as n tends to infinity, using the small subgraph conditioning method.We present several results including an asymptotic formula for the expectation of XG when G is d-regular, d ≥ 3. The interaction of perfect matchings with short cycles in random lifts of regular multigraphs is also analysed. Partial calculations are performed for the second moment of XG, with full details given for two example multigraphs, including the complete graph K4.To assist in our calculations we provide a theorem for estimating a summation over multiple dimensions using Laplace's method. This result is phrased as a summation over lattice points, and may prove useful in future applications.

Counting of spanning trees of a complete graph

2018 ◽  
Vol 40 ◽  
pp. 19
Author(s):  
Anderson Luiz Pedrosa Porto ◽  
Vagner Rodrigues de Bessa ◽  
Mattheus Pereira da Silva Aguiar ◽  
Mariana Martins Vieira
Keyword(s):  
Complete Graph ◽  
Spanning Trees

scholarly journals Perfect matchings (and Hamilton cycles) in hypergraphs with large degrees

2011 ◽  
Vol 32 (5) ◽  
pp. 677-687 ◽  
Author(s):  
Klas Markström ◽  
Andrzej Ruciński
Keyword(s):  
Perfect Matchings ◽  
Hamilton Cycles

scholarly journals Multicoloured Hamilton Cycles

10.37236/1204 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Michael Albert ◽  
Alan Frieze ◽  
Bruce Reed
Keyword(s):  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Proof Technique ◽  
Hamilton Cycles ◽  
Local Lemma

The edges of the complete graph $K_n$ are coloured so that no colour appears more than $\lceil cn\rceil$ times, where $c < 1/32$ is a constant. We show that if $n$ is sufficiently large then there is a Hamiltonian cycle in which each edge is a different colour, thereby proving a 1986 conjecture of Hahn and Thomassen. We prove a similar result for the complete digraph with $c < 1/64$. We also show, by essentially the same technique, that if $t\geq 3$, $c < (2t^2(1+t))^{-1}$, no colour appears more than $\lceil cn\rceil$ times and $t|n$ then the vertices can be partitioned into $n/t$ $t-$sets $K_1,K_2,\ldots,K_{n/t}$ such that the colours of the $n(t-1)/2$ edges contained in the $K_i$'s are distinct. The proof technique follows the lines of Erdős and Spencer's modification of the Local Lemma.

CONSTRUCTING MULTIDIMENSIONAL SPANNER GRAPHS

1991 ◽  
Vol 01 (02) ◽  
pp. 99-107 ◽  
Author(s):  
JEFFERY S. SALOWE
Keyword(s):  
Shortest Path ◽  
Complete Graph ◽  
Connected Graph ◽  
Spanning Trees ◽  
Input Parameter ◽  
Minimum Spanning Trees ◽  
Positive Edge ◽  
Spanning Subgraph ◽  
Vertex Set

Given a connected graph G=(V,E) with positive edge weights, define the distance dG(u,v) between vertices u and v to be the length of a shortest path from u to v in G. A spanning subgraph G' of G is said to be a t-spanner for G if, for every pair of vertices u and v, dG'(u,v)≤t·dG(u,v). Consider a complete graph G whose vertex set is a set of n points in [Formula: see text] and whose edge weights are given by the Lp distance between respective points. Given input parameter ∊, 0<∊≤1, we show how to construct a (1+∊)-spanner for G containing [Formula: see text] edges in [Formula: see text] time. We apply this spanner to the construction of approximate minimum spanning trees.

scholarly journals Rainbow perfect matchings and Hamilton cycles in the random geometric graph

2017 ◽  
Vol 51 (4) ◽  
pp. 587-606 ◽  
Author(s):  
Deepak Bal ◽  
Patrick Bennett ◽  
Xavier Pérez-Giménez ◽  
Paweł Prałat
Keyword(s):  
Geometric Graph ◽  
Perfect Matchings ◽  
Hamilton Cycles ◽  
Random Geometric Graph

SOME FOURTH-ORDER LINEAR DIVISIBILITY SEQUENCES

2011 ◽  
Vol 07 (05) ◽  
pp. 1255-1277 ◽  
Author(s):  
H. C. WILLIAMS ◽  
R. K. GUY
Keyword(s):  
Elliptic Curves ◽  
Large Class ◽  
Finite Fields ◽  
Fourth Order ◽  
Spanning Trees ◽  
Perfect Matchings ◽  
Second Order ◽  
Order Linear ◽  
Curves Over Finite Fields ◽  
Open Question

We extend the Lucas–Lehmer theory for second-order divisibility sequences to a large class of fourth-order sequences, with appropriate laws of apparition and of repetition. Examples are provided by the numbers of perfect matchings, or of spanning trees, in families of graphs, and by the numbers of points on elliptic curves over finite fields. Whether there are fourth-order divisibility sequences not covered by our theory is an open question.

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