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Attempting perfect hypergraphs [article]

Maria Chudnovsky, Gil Kalai
2021 arXiv   pre-print
We study several extensions of the notion of perfect graphs to k-uniform hypergraphs.  ...  We mentioned that chordal graphs have a simple homological description: G is chordal if and only if for every induced subgraph H of G, Hi(C(H)) = 0, for every i > 0, where C(H) is the clique complex of  ...  is a pre-odd-hole. • No induced subgraph of G is a pre-odd-anti-hole.  ... 
arXiv:2111.03228v1 fatcat:w5kyexh22bgtrpgas6cpqnji54

Page 7694 of Mathematical Reviews Vol. , Issue 2001K [page]

2001 Mathematical Reviews  
An odd hole is an odd chordless cycle with at least five vertices; an anti-hole is the complement of a hole.  ...  largest clique of G containing v is contained in AU B, then G is an odd hole or an odd anti-hole.  ... 

Page 2338 of Mathematical Reviews Vol. , Issue 2003d [page]

2003 Mathematical Reviews  
Namely, instead of consid- ering the hypergraph KG,(S) where edges are pairwise disjoint r-tuples of sets, the author considers ‘s’-wise disjoint r-tuples.  ...  In this paper the authors prove that a partitionable graph G is an odd hole or the complement of an odd hole, if G contains a universal 2- amalgam.  ... 

Topological Interference Management with Decoded Message Passing [article]

Xinping Yi, Giuseppe Caire
2017 arXiv   pre-print
The topological interference management (TIM) problem studies partially-connected interference networks with no channel state information except for the network topology (i.e., connectivity graph) at the  ...  For this TIM problem with decoded message passing (TIM-MP), we model the interference pattern by conflict digraphs, connect orthogonal access to the acyclic set coloring on conflict digraphs, and show  ...  When k = 2, we have r = 1, 2. So, the condition turns out to be that, if D either contains no directed odd cycle or no directed even cycle, χ A (D) ≤ 2.  ... 
arXiv:1702.08079v1 fatcat:inwwgukqlrbhhbmzxxtkrhz6r4

On three measures of non-convexity

Josef Cibulka, Miroslav Korbelář, Jan Kynčl, Viola Mészáros, Rudolf Stolař, Pavel Valtr
2017 Israel Journal of Mathematics  
As a part of the proof we show that a disc with n one-point holes near its boundary has χ(I(X)) >(n) but ω(I(X))=3. We also find sets X in R^5 with χ(X)=2, but γ(X) arbitrarily large.  ...  The invisibility graph I(X) of a set X ⊆R^d is a (possibly infinite) graph whose vertices are the points of X and two vertices are connected by an edge if and only if the straight-line segment connecting  ...  Then we show that each component with an odd cycle is 3-colorable.  ... 
doi:10.1007/s11856-017-1467-1 fatcat:bsiainkuvrcxhooc6skuartp3y

Algorithms for Perfectly Contractile Graphs

Frédéric Maffray, Nicolas Trotignon
2005 SIAM Journal on Discrete Mathematics  
class A' of graphs that contain no odd hole, no antihole of length at least 5 and no odd prism (prism whose three paths are odd).  ...  We consider the class A of graphs that contain no odd hole, no antihole of length at least 5, and no "prism" (a graph consisting of two disjoint triangles with three disjoint paths between them) and the  ...  Suppose that F contains no odd hole and R is not bipartite. Then R contains an odd cycle Z. This cycle must be a triangle, for otherwise L(R) contains an odd hole, a contradiction.  ... 
doi:10.1137/s0895480104442522 fatcat:f2owyoha35ee3cr3qt2t44xcbe

Chromatic capacities of graphs and hypergraphs

Joshua Evan Greene
2004 Discrete Mathematics  
Given a hypergraph H, the chromatic capacity cap(H) of H is the largest k for which there exists a k-coloring of the edges of H such that, for every coloring of the vertices of H with the edge colors,  ...  We also answer a question of Archer by constructing, for all r and , r-uniform hypergraphs attaining the bound cap = −1. Finally, we show that a connected graph G has cap(G)  ...  The program was supported by grants from the NSF (DMS 92820179) and the NSA (904-00-1-0026).  ... 
doi:10.1016/j.disc.2003.06.016 fatcat:gl2j5qnf3bb4nmokxramsapg6m

Graphs encoding 3-manifolds of genus 2

Alberto Cavicchioli, Fulvia Spaggiari
2008 Discrete Mathematics  
We give a simple alternative proof of the representation theorem of all genus two 3-manifolds by a 6-parameter family of integers, due to Casali and Grasselli (see [M.R.  ...  Our approach is different from that of the quoted papers, and it is based on the concept of extended Heegaard diagram.  ...  (National Research Council) of Italy.  ... 
doi:10.1016/j.disc.2006.11.044 fatcat:onsznbsabvgrjgkbfte5wya7ae

Progress on perfect graphs

Maria Chudnovsky, Neil Robertson, P. D. Seymour, Robin Thomas
2003 Mathematical programming  
cycle of length at least five, or the complement of such an odd cycle.  ...  In the second part we outline our recent proof of the Strong Perfect Graph Conjecture of Berge from 1961, the following: a graph is perfect if and only if it has no induced subgraph isomorphic to an odd  ...  Several of their ideas were seminal for this research. We would also like to acknowledge the American Institute of Mathematics, who supported two of us full-time for six months at the  ... 
doi:10.1007/s10107-003-0449-8 fatcat:a5gxizvoxnathjs3jgui6lvtiu

Clique separator decomposition of hole-free and diamond-free graphs and algorithmic consequences

Andreas Brandstädt, Vassilis Giakoumakis, Frédéric Maffray
2012 Discrete Applied Mathematics  
Clique separator decomposition, introduced by Whitesides and Tarjan, is one of the most important graph decompositions. A hole is a chordless cycle with at least five vertices.  ...  We show that every (hole, paraglider)-free graph admits a clique separator decomposition into graphs of three very specific types.  ...  Since every hole C k with k ≥ 7 contains the disjoint union of P 2 and P 3 , and the paraglider is the complementary graph of P 2 ∪ P 3 , it follows that HP-free graphs contain no odd hole and no odd antihole  ... 
doi:10.1016/j.dam.2011.10.031 fatcat:226muztrbfhyzgchj5i2q3y6xi

On rank-width of even-hole-free graphs [article]

Isolde Adler, Ngoc Khang Le, Haiko Müller, Marko Radovanović, Nicolas Trotignon, Kristina Vušković
2017 arXiv   pre-print
We present a class of (diamond, even hole)-free graphs with no clique cutset that has unbounded rank-width.  ...  Linhares-Sales (2010) showed that planar even-hole-free graphs have bounded rank-width, and N.K. Le (2016) showed that even-hole-free graphs with no star cutset have bounded rank-width.  ...  Introduction In a graph G a hole is a chordless cycle of length at least four. A hole is even or odd depending on the parity of the size of its vertex set. An n-hole is a hole on n vertices.  ... 
arXiv:1611.09907v2 fatcat:hgtsktipz5gytac7dkcjlz62le

Page 2466 of Mathematical Reviews Vol. , Issue 89E [page]

1989 Mathematical Reviews  
A k-coloring of G is a mapping f from V into the set [k] = {1,2,---,k}, and f isa proper k-coloring if no two adjacent vertices are assigned the same element of [k].  ...  Each k-coloring of G induces a partition of V(G) into k color classes.  ... 

Coloring square-free Berge graphs

Maria Chudnovsky, Irene Lo, Frédéric Maffray, Nicolas Trotignon, Kristina Vušković
2018 Journal of combinatorial theory. Series B (Print)  
We consider the class of Berge graphs that do not contain a chordless cycle of length 4.  ...  We present a purely graph-theoretical algorithm that produces an optimal coloring in polynomial time for every graph in that class.  ...  A Berge graph is any graph that contains no odd hole and no odd antihole.  ... 
doi:10.1016/j.jctb.2018.07.010 fatcat:skyww5wnpferpbhl3cjhjevcka

Coloring Square-free Berge Graphs [article]

Maria Chudnovsky, Irene Lo, Frederic Maffray, Nicolas Trotignon, Kristina Vuskovic
2015 arXiv   pre-print
We consider the class of Berge graphs that do not contain a chordless cycle of length 4.  ...  We present a purely graph-theoretical algorithm that produces an optimal coloring in polynomial time for every graph in that class.  ...  A Berge graph is any graph that contains no odd hole and no odd antihole.  ... 
arXiv:1509.09195v1 fatcat:euoxvmfbsnaqjn2zi634vn6heq

Even pairs in square-free Berge graphs with no odd prism [article]

Frédéric Maffray
2015 arXiv   pre-print
We consider the class of Berge graphs that contain no odd prism and no square (cycle on four vertices).  ...  This result is used to devise a polynomial-time algorithm to color optimally every graph in this class.  ...  A Berge graph is any graph that contains no odd hole and no odd antihole of length at least 5.  ... 
arXiv:1502.03695v1 fatcat:xo2pk6mou5gdbemtfhrg5tnjge
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