scholarly journals Multicolored isomorphic spanning trees in complete graphs

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Gregory Constantine
Keyword(s):  
Complete Graph ◽  
Spanning Trees ◽  
Complete Graphs ◽  
Edge Disjoint ◽  
International Audience

International audience Can a complete graph on an even number n (>4) of vertices be properly edge-colored with n-1 colors in such a way that the edges can be partitioned into edge disjoint colorful isomorphic spanning trees? A spanning treee is colorful if all n-1 colors occur among its edges. It is proved that this is possible to accomplish whenever n is a power of two, or five times a power of two.

scholarly journals Edge-disjoint rainbow spanning trees in complete graphs

2016 ◽  
Vol 57 ◽  
pp. 71-84 ◽  
Author(s):  
James M. Carraher ◽  
Stephen G. Hartke ◽  
Paul Horn
Keyword(s):  
Spanning Trees ◽  
Complete Graphs ◽  
Edge Disjoint

scholarly journals $Star^1$-convex functions on tropical linear spaces of complete graphs

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Laura Escobar
Keyword(s):  
Complete Graph ◽  
Piecewise Linear ◽  
Negative Answer ◽  
Linear Functions ◽  
One Dimension ◽  
Complete Graphs ◽  
Piecewise Linear Functions ◽  
Linear Spaces ◽  
International Audience ◽  
Tropical Linear Space

International audience Given a fan $\Delta$ and a cone $\sigma \in \Delta$ let $star^1(\sigma )$ be the set of cones that contain $\sigma$ and are one dimension bigger than $\sigma$ . In this paper we study two cones of piecewise linear functions defined on $\delta$ : the cone of functions which are convex on $star^1(σ\sigma)$ for all cones, and the cone of functions which are convex on $star^1(σ\sigma)$ for all cones of codimension 1. We give nice combinatorial descriptions for these two cones given two different fan structures on the tropical linear space of complete graphs. For the complete graph $K_5$, we prove that with the finer fan subdivision the two cones are not equal, but with the coarser subdivision they are the same. This gives a negative answer to a question of Gibney-Maclagan that for the finer subdivision the two cones are the same. Soit $\Delta$ un fan, pour $\sigma \in \Delta$ nous définissons $star^1(\sigma )$ comme l'ensemble de cônes qui contiennent $\sigma$ dont la dimension est un de plus que la dimension de $\sigma$ . Nous étudions deux cônes d'applications linéaires par morceaux définis sur $\Delta$ : le cône de fonctions convexes sur$star^1(\sigma )$, où $\sigma \in \Delta$ est un cône quelconque, et le cône de fonctions convexes sur $star^1(σ\sigma)$ où σ est un cône de codimension 1. étant donnés deux structures sur l'espace tropical linéaire de graphes complets, nous donnons de beaux descriptions combinatoires des cônes décrits en haut. Pour le graphe complet $K_5$, on démontre que avec la subdivision en fans plus fine, les deux cônes sont différentes, mais avec la subdivision plus gros ils sont cônes sont les mêmes. Ce résultant réponde négativement une question de Gibney-Maclagan.

scholarly journals Toppling numbers of complete and random graphs

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Anthony Bonato ◽  
William B. Kinnersley ◽  
Pawel Pralat
Keyword(s):  
Graph Theory ◽  
Differential Equations ◽  
Random Graphs ◽  
Complete Graph ◽  
Complete Graphs ◽  
Asymptotic Bounds ◽  
Large N ◽  
Chip Firing ◽  
International Audience ◽  
Person Game

Graph Theory International audience We study a two-person game played on graphs based on the widely studied chip-firing game. Players Max and Min alternately place chips on the vertices of a graph. When a vertex accumulates as many chips as its degree, it fires, sending one chip to each neighbour; this may in turn cause other vertices to fire. The game ends when vertices continue firing forever. Min seeks to minimize the number of chips played during the game, while Max seeks to maximize it. When both players play optimally, the length of the game is the toppling number of a graph G, and is denoted by t(G). By considering strategies for both players and investigating the evolution of the game with differential equations, we provide asymptotic bounds on the toppling number of the complete graph. In particular, we prove that for sufficiently large n 0.596400 n2 < t(Kn) < 0.637152 n2. Using a fractional version of the game, we couple the toppling numbers of complete graphs and the binomial random graph G(n,p). It is shown that for pn ≥n² / √ log(n) asymptotically almost surely t(G(n,p))=(1+o(1)) p t(Kn).

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 $G$.

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. . . .

scholarly journals On the ranks of configurations on the complete graph

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Robert Cori ◽  
Yvan Le Borgne
Keyword(s):  
Complete Graph ◽  
Efficient Algorithm ◽  
Complete Graphs ◽  
International Audience ◽  
Definition Of ◽  
Dyck Words

International audience We consider the parameter rank introduced for graph configurations by M. Baker and S. Norine. We focus on complete graphs and obtain an efficient algorithm to determine the rank for these graphs. The analysis of this algorithm leads to the definition of a parameter on Dyck words, which we call prerank. We prove that the distribution of area and prerank on Dyck words of given length $2n$ leads to a polynomial with variables $q,t$ which is symmetric in these variables. This polynomial is different from the $q,t-$Catalan polynomial studied by A. Garsia, J. Haglund and M. Haiman.

scholarly journals Decomposition of complete graphs into connected unicyclic graphs with eight edges and pentagon

2019 ◽  
Vol 3 (1) ◽  
pp. 24
Author(s):  
Dalibor Froncek ◽  
O'Neill Kingston
Keyword(s):  
Complete Graph ◽  
Necessary Conditions ◽  
Complete Graphs ◽  
Decomposition Techniques ◽  
Unicyclic Graphs ◽  
Edge Disjoint

<p>A <span class="math"><em>G</em></span>-decomposition of the complete graph <span class="math"><em>K</em><sub><em>n</em></sub></span> is a family of pairwise edge disjoint subgraphs of <span class="math"><em>K</em><sub><em>n</em></sub></span>, all isomorphic to <span class="math"><em>G</em></span>, such that every edge of <span class="math"><em>K</em><sub><em>n</em></sub></span> belongs to exactly one copy of <span class="math"><em>G</em></span>. Using standard decomposition techniques based on <span class="math"><em>ρ</em></span>-labelings, introduced by Rosa in 1967, and their modifications we show that each of the ten non-isomorphic connected unicyclic graphs with eight edges containing the pentagon decomposes the complete graph <span class="math"><em>K</em><sub><em>n</em></sub></span> whenever the necessary conditions are satisfied.</p>

scholarly journals A chip-firing variation and a new proof of Cayley's formula

2013 ◽  
Vol Vol. 15 no. 1 (Graph Theory) ◽  
Author(s):  
Peter Mark Kayll ◽  
Dave Perkins
Keyword(s):  
Graph Theory ◽  
Spanning Trees ◽  
Complete Graphs ◽  
Physical Processes ◽  
Connected Graphs ◽  
Bijective Proof ◽  
Chip Firing ◽  
International Audience ◽  
General Graphs

Graph Theory International audience We introduce a variation of chip-firing games on connected graphs. These 'burn-off' games incorporate the loss of energy that may occur in the physical processes that classical chip-firing games have been used to model. For a graph G=(V,E), a configuration of 'chips' on its nodes is a mapping C:V→ℕ. We study the configurations that can arise in the course of iterating a burn-off game. After characterizing the 'relaxed legal' configurations for general graphs, we enumerate the 'legal' ones for complete graphs Kn. The number of relaxed legal configurations on Kn coincides with the number tn+1 of spanning trees of Kn+1. Since our algorithmic, bijective proof of this fact does not invoke Cayley's Formula for tn, our main results yield secondarily a new proof of this formula.

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 A proof of Ringel’s conjecture

2021 ◽  
Author(s):  
R. Montgomery ◽  
A. Pokrovskiy ◽  
B. Sudakov
Keyword(s):  
Complete Graph ◽  
Complete Graphs ◽  
Edge Disjoint ◽  
Ringel’S Conjecture

AbstractA typical decomposition question asks whether the edges of some graph G can be partitioned into disjoint copies of another graph H. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with n edges packs $$2n+1$$ 2 n + 1 times into the complete graph $$K_{2n+1}$$ K 2 n + 1 . In this paper, we prove this conjecture for large n.

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