linear hypergraphs
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2022 ◽  
Vol 99 ◽  
pp. 103401
Author(s):  
Emma Cohen ◽  
Will Perkins ◽  
Michail Sarantis ◽  
Prasad Tetali

Author(s):  
Oliver Janzer ◽  
Zoltán Lóránt Nagy

AbstractThe long-standing Erdős–Faber–Lovász conjecture states that every n-uniform linear hypergaph with n edges has a proper vertex-coloring using n colors. In this paper we propose an algebraic framework to the problem and formulate a corresponding stronger conjecture. Using the Combinatorial Nullstellensatz, we reduce the Erdős–Faber–Lovász conjecture to the existence of non-zero coefficients in certain polynomials. These coefficients are in turn related to the number of orientations with prescribed in-degree sequences of some auxiliary graphs. We prove the existence of certain orientations, which verifies a necessary condition for our algebraic approach to work.


2021 ◽  
Vol 37 (3) ◽  
pp. 867-890
Author(s):  
Michael A. Henning ◽  
Anders Yeo

2020 ◽  
Vol 30 (03) ◽  
pp. 2040003
Author(s):  
Shangwei Lin ◽  
Jianfeng Pei ◽  
Chunfang Li

A connected graph [Formula: see text] is super edge-connected, if every minimum edge-cut of [Formula: see text] is the set of edges incident with a vertex. In this paper, the concept of super edge-connectivity of graphs is generalized to hypergraphs and a necessary and sufficient condition for an [Formula: see text]-uniform and linear hypergraph with diameter at most 2 to be super edge-connected is given.


2020 ◽  
Vol 29 (5) ◽  
pp. 698-721
Author(s):  
Tao Jiang ◽  
Liana Yepremyan

AbstractA classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in an n-vertex graph not containing C2k, the cycle of length 2k, is O(n1+1/k). Simonovits established a corresponding supersaturation result for C2k’s, showing that there exist positive constants C,c depending only on k such that every n-vertex graph G with e(G)⩾ Cn1+1/k contains at least c(e(G)/v(G))2k copies of C2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant).In this paper we extend Simonovits' result to a supersaturation result of r-uniform linear cycles of even length in r-uniform linear hypergraphs. Our proof is self-contained and includes the r = 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.


2019 ◽  
Vol 93 (1) ◽  
pp. 113-141
Author(s):  
József Balogh ◽  
Lina Li

2019 ◽  
Vol 163 ◽  
pp. 163-181 ◽  
Author(s):  
Beka Ergemlidze ◽  
Ervin Győri ◽  
Abhishek Methuku
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