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

1997
*
Journal of combinatorial theory. Series A
*

Let H be

doi:10.1006/jcta.1996.2728
fatcat:zzddbl7qfrdoxhxwf45zgt4cfm
*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 ...##
###
Graph and hypergraph colouring via nibble methods: A survey
[article]

2021
*
arXiv
*
pre-print

This paper provides

arXiv:2106.13733v3
fatcat:5apaxr4x7vanbofnzdzjhnhsb4
*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 ...##
###
On the Chromatic Thresholds of Hypergraphs

2015
*
Combinatorics, probability & computing
*

Let$\mathcal{F}$be

doi:10.1017/s0963548315000061
fatcat:bi6h4clqbfd4fa3johomw65p4y
*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. ...##
###
List Coloring Hypergraphs

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. ...

##
###
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

1992
*
Journal of combinatorial theory. Series A
*

It is shown that

doi:10.1016/0097-3165(92)90096-d
fatcat:fnhyemvrandxjhivekp6z5564e
*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. ...##
###
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]

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. ...

##
###
A Note on Embedding Hypertrees

2009
*
Electronic Journal of Combinatorics
*

An $r$-tree is

doi:10.37236/256
fatcat:5yzhe76tofgxrdbwe2irucux4u
*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. ...##
###
Asymptotics of Hypergraph Matching, Covering and Coloring Problems
[chapter]

1995
*
Proceedings of the International Congress of Mathematicians
*

This at first seems unpromising, because wc are accustomed to thinking

doi:10.1007/978-3-0348-9078-6_130
fatcat:34psrgyyv5amzbilfkjrfykerq
*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. ...##
###
The Irreducible Characters for Semi-Simple Algebraic Groups and for Quantum Groups
[chapter]

1995
*
Proceedings of the International Congress of Mathematicians
*

This at first seems unpromising, because wc are accustomed to thinking

doi:10.1007/978-3-0348-9078-6_66
fatcat:zwbcvzbwibc55igkkponkbd2n4
*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. ...##
###
Asymptotics of the Chromatic Index for Multigraphs

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*. ...

##
###
Plurigraph coloring and scheduling problems
[article]

2017
*
arXiv
*
pre-print

Also, we show that

arXiv:1611.09469v2
fatcat:jnsfpnsdobestjvvlf2q7vhyhq
*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. ...##
###
On_the_coloring_of_graphs_formed_by_cliques_sharing_atmost_one_common_point.pdf
[article]

2019
*
Figshare
*

In this work, we try to prove that

doi:10.6084/m9.figshare.8306441.v1
fatcat:6anx4mr7e5hy3fu3yaiswbdwii
*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. ...##
###
Hedetniemi's conjecture from the topological viewpoint
[article]

2019
*
arXiv
*
pre-print

We fully confirm

arXiv:1806.04963v3
fatcat:4fjo4u2pbzbf3a3blij6kczbf4
*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. ...
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