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On the Degree, Size, and Chromatic Index of a Uniform Hypergraph

Noga Alon, Jeong Han Kim
1997 Journal of combinatorial theory. Series A  
Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the maximum degree of a vertex of H.  ...  We conjecture that for every > 0, if D is sufficiently large as a function of t, k and , then the chromatic index of H is at most (t − 1 + 1/t + )D.  ...  Introduction For a k-uniform hypergraph H (which may have multiple edges), let D(H) denote the maximum degree of a vertex of H, and let χ (H) denote the chromatic index of H, that is, the minimum number  ... 
doi:10.1006/jcta.1996.2728 fatcat:zzddbl7qfrdoxhxwf45zgt4cfm

Graph and hypergraph colouring via nibble methods: A survey [article]

Dong Yeap Kang, Tom Kelly, Daniela Kühn, Abhishek Methuku, Deryk Osthus
2021 arXiv   pre-print
This paper provides a survey of methods, results, and open problems on graph and hypergraph colourings, with a particular emphasis on semi-random 'nibble' methods.  ...  We also give a detailed sketch of some aspects of the recent proof of the Erdős-Faber-Lovász conjecture.  ...  Acknowledgements We are grateful to Jeff Kahn for helpful comments on a preliminary version of the survey and to Dhruv Mubayi for informing us that the conjecture of Frieze and Mubayi [61] listed in  ... 
arXiv:2106.13733v3 fatcat:5apaxr4x7vanbofnzdzjhnhsb4

On the Chromatic Thresholds of Hypergraphs

JÓZSEF BALOGH, JANE BUTTERFIELD, PING HU, JOHN LENZ, DHRUV MUBAYI
2015 Combinatorics, probability & computing  
Let$\mathcal{F}$be a family ofr-uniform hypergraphs.  ...  In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing  ...  Let A be an r-uniform hypergraph. Our method of proving an upper bound on the chromatic threshold of the family of A-free hypergraphs, used in Theorems 1 and 7, is the following.  ... 
doi:10.1017/s0963548315000061 fatcat:bi6h4clqbfd4fa3johomw65p4y

List Coloring Hypergraphs

Penny Haxell, Jacques Verstraete
2010 Electronic Journal of Combinatorics  
On the other hand there exist $d$-regular three-uniform linear hypergraphs with list chromatic number at most $\log_3 d+3$.  ...  The hypergraph $H$ is $k$-list-colorable if it has a list coloring from any collection of lists of size $k$. The list chromatic number of $H$ is the minimum $k$ such that $H$ is $k$-list-colorable.  ...  Acknowledgments This work was done while the authors were visiting the Institute for Pure and Applied Mathematics at the University of California, Los Angeles.  ... 
doi:10.37236/401 fatcat:b7t7lhziuzfldn4la5ixnorlsm

Page 7341 of Mathematical Reviews Vol. , Issue 97M [page]

1997 Mathematical Reviews  
{For the entire collection see MR 97i:68012.} 97m:05190 05C65 Alon, Noga (IL-TLAV-DM; Tel Aviv); Kim, Jeong Han (1-BELL-M; Murray Hill, NJ) On the degree, size, and chromatic index of a uniform hypergraph  ...  Let # be a k-uniform hypergraph in which no two edges share more than f common vertices, let D denote the maximum degree of a vertex of #, and let 7’ denote the chromatic index of #, that is, the minimum  ... 

Coloring nearly-disjoint hypergraphs with n + o(n) colors

Jeff Kahn
1992 Journal of combinatorial theory. Series A  
It is shown that the chromatic index of a nearly-disjoint hypergraph on n vertices is at most n + o(n).  ...  This is an approximate version of the well-known conjecture of Erdiis, Faber, and Lo&z stating that the chromatic index is at most n. 0 1992 Academic Press, Inc.  ...  A matching of 2 is a collection of pairwise disjoint edges, and the chromatic index, f(X), of X is the least size of a collection of matchings whose union is X.  ... 
doi:10.1016/0097-3165(92)90096-d fatcat:fnhyemvrandxjhivekp6z5564e

Page 8941 of Mathematical Reviews Vol. , Issue 2003m [page]

2003 Mathematical Reviews  
Summary: “A graph is said to be of class I if its chromatic index is equal to the maximum degree of this graph.  ...  The stability number a(H) of a hypergraph H is the cardinality of the largest set of vertices of H which does not contain an edge. A hypergraph is k-uniform if the sizes of ail its edges are k.  ... 

A note on embedding hypertrees [article]

Po-Shen Loh
2009 arXiv   pre-print
A classical result from graph theory is that every graph with chromatic number χ > t contains a subgraph with all degrees at least t, and therefore contains a copy of every t-edge tree.  ...  An r-tree is an r-uniform hypergraph with no pair of edges intersecting in more than one vertex, and no sequence of distinct vertices and edges (v_1, e_1, ..., v_k, e_k) with all e_i ∋v_i, v_i+1, where  ...  Also, he thanks Asaf Shapira for pointing out the application of the main theorem to Chvátal's result, and the referee for carefully reading this article.  ... 
arXiv:0901.2988v3 fatcat:q4mevv5s5za4heobv7a65vqf6u

A Note on Embedding Hypertrees

Po-Shen Loh
2009 Electronic Journal of Combinatorics  
An $r$-tree is a connected $r$-uniform hypergraph with no pair of edges intersecting in more than one vertex, and no sequence of distinct vertices and edges $(v_1, e_1, \ldots, v_k, e_k)$ with all $e_i  ...  A classical result from graph theory is that every graph with chromatic number $\chi > t$ contains a subgraph with all degrees at least $t$, and therefore contains a copy of every $t$-edge tree.  ...  The author thanks his Ph.D. advisor, Benny Sudakov, for introducing him to this problem, and for remarks that helped to improve the exposition of this note.  ... 
doi:10.37236/256 fatcat:5yzhe76tofgxrdbwe2irucux4u

Asymptotics of Hypergraph Matching, Covering and Coloring Problems [chapter]

Jeff Kahn
1995 Proceedings of the International Congress of Mathematicians  
This at first seems unpromising, because wc are accustomed to thinking of the degree as a trivial lower bound on chromatic index.  ...  List-colorings The list-chromatic index, x|(W), of H is the least t such that if S (A) is a set ("list") of size t for each A E H, then there exists a coloring a of H with o~(A) G S(A) for each A G H.  ... 
doi:10.1007/978-3-0348-9078-6_130 fatcat:34psrgyyv5amzbilfkjrfykerq

The Irreducible Characters for Semi-Simple Algebraic Groups and for Quantum Groups [chapter]

Henning Haahr Andersen
1995 Proceedings of the International Congress of Mathematicians  
This at first seems unpromising, because wc are accustomed to thinking of the degree as a trivial lower bound on chromatic index.  ...  List-colorings The list-chromatic index, x|(W), of H is the least t such that if S (A) is a set ("list") of size t for each A E H, then there exists a coloring a of H with o~(A) G S(A) for each A G H.  ... 
doi:10.1007/978-3-0348-9078-6_66 fatcat:zwbcvzbwibc55igkkponkbd2n4

Asymptotics of the Chromatic Index for Multigraphs

Jeff Kahn
1996 Journal of combinatorial theory. Series B (Print)  
The chromatic index, /$(G), of G is the least size of a 1 admitting such a coloring. We write D(G) for the maximum degree of G. For further terminology see below.  ...  A 2-uniform hypergraph is a multigraph and a simple multigraph is a graph. For further hypergraph background see e.g. [15] or [7] . Fractional chromatic index.  ... 
doi:10.1006/jctb.1996.0067 fatcat:xg5532nxc5hvzipgmm2qla4wke

Plurigraph coloring and scheduling problems [article]

John Machacek
2017 arXiv   pre-print
Also, we show that the chromatic symmetric function determines the degree sequence of uniform hypertrees, but there exist pairs on 3-uniform hypertrees which are not isomorphic yet have the same chromatic  ...  We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes.  ...  Acknowledgments The author wishes to thank the anonymous reviewer for his or her reading and helpful suggestions.  ... 
arXiv:1611.09469v2 fatcat:jnsfpnsdobestjvvlf2q7vhyhq

On_the_coloring_of_graphs_formed_by_cliques_sharing_atmost_one_common_point.pdf [article]

Ganitarthi
2019 Figshare  
In this work, we try to prove that the chromatic number of the graph formed by adjoining k cliques of order k, any two of which meet at a single vertex is k  ...  [4] showed that for any simple hypergraph H on n vertices, the chromatic index of H is at most [1.5n-2] . Kahn [3] proved that the chromatic number of H is at most n + o(n). Jackson et al.  ...  Faber [5] proved that for fixed degree, there can be only finitely many counterexamples to EFL on both regular and uniform hypergraphs. Hegde et al.  ... 
doi:10.6084/m9.figshare.8306441.v1 fatcat:6anx4mr7e5hy3fu3yaiswbdwii

Hedetniemi's conjecture from the topological viewpoint [article]

Hamid Reza Daneshpajouh, Roman Karasev, Alexey Yu. Volovikov
2019 arXiv   pre-print
We fully confirm the version of this conjecture for the homological index via establishing a stronger formula for the homological index of the join of Z/2-spaces.  ...  This paper is devoted to studying a topological version of the famous Hedetniemi conjecture which says: The Z/2-index of the Cartesian product of two Z/2-spaces is equal to the minimum of their Z/2-indexes  ...  An r-uniform hypergraph is a hypergraph such that all its edges have size r. A 2-uniform hypergraph is simply called a graph.  ... 
arXiv:1806.04963v3 fatcat:4fjo4u2pbzbf3a3blij6kczbf4
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