scholarly journals Graph Orientations and Linear Extensions.

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Benjamin Iriarte
Keyword(s):  
Partial Order ◽  
Simple Graph ◽  
Linear Extensions ◽  
Acyclic Orientations ◽  
International Audience ◽  
General Graphs ◽  
Odd Cycles

International audience Given an underlying undirected simple graph, we consider the set of all acyclic orientations of its edges. Each of these orientations induces a partial order on the vertices of our graph, and therefore we can count the number of linear extensions of these posets. We want to know which choice of orientation maximizes the number of linear extensions of the corresponding poset, and this problem is solved essentially for comparability graphs and odd cycles, presenting several proofs. We then provide an inequality for general graphs and discuss further techniques.

scholarly journals A note on compact and compact circular edge-colorings of graphs

2008 ◽  
Vol Vol. 10 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Dariusz Dereniowski ◽  
Adam Nadolski
Keyword(s):  
Approximation Algorithm ◽  
Decision Problem ◽  
Edge Coloring ◽  
Exact Algorithm ◽  
Weighted Graphs ◽  
Edge Colorings ◽  
Odd Cycle ◽  
International Audience ◽  
Np Complete ◽  
Odd Cycles

Graphs and Algorithms International audience We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.

scholarly journals Operations on partially ordered sets and rational identities of type A

2013 ◽  
Vol Vol. 15 no. 2 (Combinatorics) ◽  
Author(s):  
Adrien Boussicault
Keyword(s):  
Partially Ordered Sets ◽  
Rational Functions ◽  
Hasse Diagram ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Closed Expression ◽  
Schubert Polynomials ◽  
Special Cases ◽  
The Family ◽  
International Audience

Combinatorics International audience We consider the family of rational functions ψw= ∏( xwi - xwi+1 )-1 indexed by words with no repetition. We study the combinatorics of the sums ΨP of the functions ψw when w describes the linear extensions of a given poset P. In particular, we point out the connexions between some transformations on posets and elementary operations on the fraction ΨP. We prove that the denominator of ΨP has a closed expression in terms of the Hasse diagram of P, and we compute its numerator in some special cases. We show that the computation of ΨP can be reduced to the case of bipartite posets. Finally, we compute the numerators associated to some special bipartite graphs as Schubert polynomials.

Improving the Predicting Power of Partial Order Based QSARs through Linear Extensions

2002 ◽  
Vol 42 (4) ◽  
pp. 806-811 ◽  
Author(s):  
Lars Carlsen ◽  
Dorte B. Lerche ◽  
Peter B. Sørensen
Keyword(s):  
Partial Order ◽  
Linear Extensions

scholarly journals The Hodge Structure of the Coloring Complex of a Hypergraph (Extended Abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Sarah C Rundell ◽  
Jane H Long
Keyword(s):  
Euler Characteristic ◽  
Homology Group ◽  
Hodge Structure ◽  
Hodge Decomposition ◽  
Simple Graph ◽  
Absolute Value ◽  
International Audience ◽  
The Absolute ◽  
Nous Examinons ◽  
Coloring Complex

International audience Let $G$ be a simple graph with $n$ vertices. The coloring complex$ Δ (G)$ was defined by Steingrímsson, and the homology of $Δ (G)$ was shown to be nonzero only in dimension $n-3$ by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group $H_{n-3}(Δ (G))$ where the dimension of the $j^th$ component in the decomposition, $H_{n-3}^{(j)}(Δ (G))$, equals the absolute value of the coefficient of $λ ^j$ in the chromatic polynomial of $G, _{\mathcal{χg}}(λ )$. Let $H$ be a hypergraph with $n$ vertices. In this paper, we define the coloring complex of a hypergraph, $Δ (H)$, and show that the coefficient of $λ ^j$ in $χ _H(λ )$ gives the Euler Characteristic of the $j^{th}$ Hodge subcomplex of the Hodge decomposition of $Δ (H)$. We also examine conditions on a hypergraph, $H$, for which its Hodge subcomplexes are Cohen-Macaulay, and thus where the absolute value of the coefficient of $λ ^j$ in $χ _H(λ )$ equals the dimension of the $j^{th}$ Hodge piece of the Hodge decomposition of $Δ (H)$. Soit $G$ un graphe simple à n sommets. Le complexe de coloriage $Δ (G)$ a été défini par Steingrímsson et Jonsson a prouvé que l'homologie de $Δ (G)$ est non nulle seulement en dimension $n-3$. Hanlon a récemment prouvé que les idempotents eulériens fournissent une décomposition du groupe d'homologie $H_{n-3}(Δ (G))$ où la dimension de la $j^e$ composante dans la décomposition de $H_{n-3}^{(j)}(Δ (G))$ est égale à la valeur absolue du coefficient de $λ ^j$ dans le polynôme chromatique de $G, _{\mathcal{χg}}(λ )$ . Soit H un hypergraphe à $n$ sommets. Dans ce texte, nous définissons le complexe de coloration d'un hypergraphe $Δ (H)$ et nous prouvons que le coefficient de $λ ^j$ dans $χ _H(λ )$ donne la caractéristique d'Euler du $j^e$ sous-complexe de Hodge dans la décomposition de Hodge de Δ (H). Nous examinons également des conditions sur un hypergraphe H pour lesquelles les sous-complexes de Hodge sont Cohen-Macaulay. Ainsi la valeur absolue du coefficient de $λ ^j$ in $χ _H(λ )$ est égale à la dimension du $j^e$sous-complexe de Hodge dans la décomposition de Hodge de $Δ (H)$.

scholarly journals Complexity results on graphs with few cliques

2007 ◽  
Vol Vol. 9 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Bill Rosgen ◽  
Lorna Stewart
Keyword(s):  
Maximum Clique ◽  
Maximum Clique Problem ◽  
Polynomial Time Algorithms ◽  
Helly Property ◽  
Graph Class ◽  
International Audience ◽  
Edge Clique Cover ◽  
Np Complete ◽  
General Graphs ◽  
Clique Problem

Graphs and Algorithms International audience A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs, some problems that are NP-complete on general graphs, such as the maximum clique problem and the maximum weighted clique problem, admit polynomial time algorithms. Other problems, such as the vertex clique cover and edge clique cover problems remain NP-complete on these classes. Several classes of graphs which have few cliques are discussed, and the complexity of some partitioning and covering problems are determined for the class of all graphs which have fewer cliques than a given polynomial bound.

scholarly journals An algorithm which generates linear extensions for a generalized Young diagram with uniform probability

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Kento Nakada ◽  
Shuji Okamura
Keyword(s):  
Young Diagram ◽  
Linear Extensions ◽  
Reduced Decompositions ◽  
Uniform Probability ◽  
Hook Formula ◽  
International Audience

International audience The purpose of this paper is to present an algorithm which generates linear extensions for a generalized Young diagram, in the sense of D. Peterson and R. A. Proctor, with uniform probability. This gives a proof of a D. Peterson's hook formula for the number of reduced decompositions of a given minuscule elements. \par Le but de ce papier est présenter un algorithme qui produit des extensions linéaires pour un Young diagramme généralisé dans le sens de D. Peterson et R. A. Proctor, avec probabilité constante. Cela donne une preuve de la hook formule d'un D. Peterson pour le nombre de décompositions réduites d'un éléments minuscules donné.

scholarly journals On the k-restricted structure ratio in planar and outerplanar graphs

2008 ◽  
Vol Vol. 10 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Gruia Călinescu ◽  
Cristina G. Fernandes
Keyword(s):  
Approximation Algorithms ◽  
Planar Graphs ◽  
Rates Of Convergence ◽  
Simple Graph ◽  
Maximum Weight ◽  
Outerplanar Graphs ◽  
Weighted Version ◽  
International Audience ◽  
Unweighted Case

Graphs and Algorithms International audience A planar k-restricted structure is a simple graph whose blocks are planar and each has at most k vertices. Planar k-restricted structures are used by approximation algorithms for Maximum Weight Planar Subgraph, which motivates this work. The planar k-restricted ratio is the infimum, over simple planar graphs H, of the ratio of the number of edges in a maximum k-restricted structure subgraph of H to the number edges of H. We prove that, as k tends to infinity, the planar k-restricted ratio tends to 1 = 2. The same result holds for the weighted version. Our results are based on analyzing the analogous ratios for outerplanar and weighted outerplanar graphs. Here both ratios tend to 1 as k goes to infinity, and we provide good estimates of the rates of convergence, showing that they differ in the weighted from the unweighted case.

scholarly journals An extremal problem for a graphic sequence to have a realization containing every 2-tree with prescribed size

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
De-Yan Zeng ◽  
Jian-Hua Yin
Keyword(s):  
Lower Bound ◽  
Extremal Problem ◽  
Simple Graph ◽  
Graphic Sequence ◽  
International Audience

International audience A graph $G$ is a $2$<i>-tree</i> if $G=K_3$, or $G$ has a vertex $v$ of degree 2, whose neighbors are adjacent, and $G-v$ is a 2-tree. Clearly, if $G$ is a 2-tree on $n$ vertices, then $|E(G)|=2n-3$. A non-increasing sequence $\pi =(d_1, \ldots ,d_n)$ of nonnegative integers is a <i>graphic sequence</i> if it is realizable by a simple graph $G$ on $n$ vertices. Yin and Li (Acta Mathematica Sinica, English Series, 25(2009)795&#x2013;802) proved that if $k \geq 2$, $n \geq \frac{9}{2}k^2 + \frac{19}{2}k$ and $\pi =(d_1, \ldots ,d_n)$ is a graphic sequence with $\sum \limits_{i=1}^n d_i > (k-2)n$, then $\pi$ has a realization containing every tree on $k$ vertices as a subgraph. Moreover, the lower bound $(k-2)n$ is the best possible. This is a variation of a conjecture due to Erd&#x0151;s and S&oacute;s. In this paper, we investigate an analogue extremal problem for 2-trees and prove that if $k \geq 3$, $n \geq 2k^2-k$ and $\pi =(d_1, \ldots ,d_n)$ is a graphic sequence with $\sum \limits_{i=1}^n d_i > \frac{4kn}{3} - \frac{5n}{3}$ then $\pi$ has a realization containing every 2-tree on $k$ vertices as a subgraph. We also show that the lower bound $\frac{4kn}{3} - \frac{5n}{3}$ is almost the best possible.

scholarly journals A variant of Niessen’s problem on degreesequences of graphs

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Jiyun Guo ◽  
Jianhua Yin
Keyword(s):  
Graph Theory ◽  
Discrete Math ◽  
Simple Graph ◽  
The Other ◽  
Degree Sequences ◽  
International Audience

Graph Theory International audience Let (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) be two sequences of nonnegative integers satisfying the condition that b1>=b2>=...>=bn, ai<= bi for i=1,2,\textellipsis,n and ai+bi>=ai+1+bi+1 for i=1,2,\textellipsis, n-1. In this paper, we give two different conditions, one of which is sufficient and the other one necessary, for the sequences (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) such that for every (c1,c2,\textellipsis,cn) with ai<=ci<=bi for i=1,2,\textellipsis,n and &#x2211;&limits;i=1n ci=0 (mod 2), there exists a simple graph G with vertices v1,v2,\textellipsis,vn such that dG(vi)=ci for i=1,2,\textellipsis,n. This is a variant of Niessen\textquoterights problem on degree sequences of graphs (Discrete Math., 191 (1998), 247&#x2013;253).

scholarly journals Packing triangles in low degree graphs and indifference graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Gordana Manić ◽  
Yoshiko Wakabayashi
Keyword(s):  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Simple Graph ◽  
Set Packing ◽  
The Best Approximation ◽  
Low Degree ◽  
International Audience ◽  
Approximation Guarantee ◽  
Vertex Disjoint

International audience We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation guarantee known so far for these problems has ratio $3/2 + ɛ$, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver in 1989. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs.

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