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Generalized k-tuple colorings of cycles and other graphs

1982
*
Journal of combinatorial theory. Series B (Print)
*

Stahl's concept

doi:10.1016/0095-8956(82)90079-x
fatcat:iac3kk5jrndy5mgphknpmxumjm
*of**k*-*tuple**coloring*the nodes*of*a*graph*is extended to specify that adjacent nodes must have i*colors*in common. ... Complete results for a minimum such*coloring*are obtained for bipartite*graphs**and*odd*cycles*. Partial results are shown for complete*graphs*. ... Stahl [7] introduced a "*k*-*tuple**coloring*"*of*a*graph*G as an assignment*of**k**colors*to each node*of*G in such a way that adjacent nodes are assigned distinct*colors*. ...##
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n-Tuple colorings and associated graphs

1976
*
Journal of combinatorial theory. Series B (Print)
*

In G,
Figure 4 contains 1 -

doi:10.1016/0095-8956(76)90010-1
fatcat:2adooxba6jchve5xdceggzvakm
*tuple**and*2-*tuple**colorings**of*a*graph*G, so x1(G) < 3*and*x2(G) < 5. Since G contains a 5-*cycle*we must have x%(G) 3 xn(C5) = 2n + 1 + [(n -1)/2]. ... On the*other*hand, 6 below show examples*of*minimal I-*tuple**and*2-*tuple**colorings**of*C, . The*color*classes in Fig. 5 consist*of*two maximal independent subsets*and*a singleton. ...##
###
Neighbor-distinguishing k-tuple edge-colorings of graphs

2009
*
Discrete Mathematics
*

This paper studies proper

doi:10.1016/j.disc.2009.04.003
fatcat:lh3vyu2cmfa2fdxouvruwldcl4
*k*-*tuple*edge-*colorings**of**graphs*that distinguish neighboring vertices by their sets*of**colors*. ... Minimum numbers*of**colors*for such*colorings*are determined for*cycles*, complete*graphs**and*complete bipartite*graphs*. ... Section 2 presents some*general*simple results about the*k*-*tuple*ND-chromatic*and*cyclic*k*-*tuple*ND-chromatic indices*of**graphs*. ...##
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Packing Loose Hamilton Cycles
[article]

2016
*
arXiv
*
pre-print

A subset C

arXiv:1608.01278v1
fatcat:ew37ldog5zfozjzpkdaa5ydjei
*of*edges in a*k*-uniform hypergraph H is a loose Hamilton*cycle*if C covers all the vertices*of*H*and*there exists a cyclic ordering*of*these vertices such that the edges in C are segments*of*... The binomial random*k*-uniform hypergraph H^k_n,p has vertex set [n]*and*an edge set E obtained by adding each*k*-*tuple*e∈[n]*k*to E with probability p, independently at random. ... Acknowledgements The authors would like to thank the anonymous referees for their very helpful comments*and*suggestions. ...##
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On k-tuple and k-tuple total domination numbers of regular graphs
[article]

2018
*
arXiv
*
pre-print

Henning

arXiv:1709.01245v2
fatcat:k645vqca2bb2pg6tyoqmh3ttfe
*and*Yeo in hen proved that if G is a cubic*graph*different from the Heawood*graph*, γ_× 2, t(G) ≤5/6n,*and*this bound is sharp. ... The*k*-*tuple*domination number*of*G, denoted by γ_×*k*(G), is the minimum cardinality*of*a*k*-*tuple*dominating set*of*G. ... In [9] , Klasing*and*Laforest described a (ln |V | + 1)-approximation algorithm for the*k*-*tuple*domination problem in*general**graphs*,*and*showed that*k*-*tuple*domination cannot be approximated within a ...##
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On the Combinatorial Power of the Weisfeiler-Lehman Algorithm
[article]

2017
*
arXiv
*
pre-print

We focus on two fundamental invariants, the num- ber

arXiv:1704.01023v1
fatcat:g2j4vvuccvejdd6bu3xawb6csm
*of**cycles*Cp*of*length p,*and*the number*of*cliques Kp*of*size p. ... The classical Weisfeiler-Lehman method WL[2] uses edge*colors*to produce a powerful*graph*invariant. ... For*k*≥ 2 every*k*-*tuple*(*and*thus also every vertex) knows the multiset*of**colors**of*all*k*-*tuples**of*vertices*of*the same*graph*. Note that the result does not hold for*k*= 1. ...##
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Multicoloring and Mycielski construction

2008
*
Discrete Mathematics
*

We then investigate the kth chromatic number

doi:10.1016/j.disc.2007.07.015
fatcat:mqlgnwarvzfargyfjcv2agqzri
*of*Mycielskians*of**cycles**and*determine the kth chromatic number*of*p-Mycielskian*of*a complete*graph**K*n for any integers*k*1, p 0*and*n 2. ... In this paper, we study the kth chromatic numbers*k**of*Mycielskians*and**generalized*Mycielskians*of**graphs*. ... On the*other*hand, if h is any proper*k*-*tuple**coloring**of*(G), let*k*0 be a*color*in h(u), we define a*k*-*tuple**coloring*c*of*G as follows: for each j = 1, 2, . . . , n, let c(v 0 j ) = h(v 0 j ) if*k*0 ...##
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The Combinatorics of Real Double Hurwitz Numbers with Real Positive Branch Points

2015
*
International mathematics research notices
*

us to uncover the beautiful combinatorics

doi:10.1093/imrn/rnv135
fatcat:aaddkb23zrddpksxfnmntbvolq
*of*these numbers both in tropical geometry*and*in the Cayley*graph*. ... Second, we express the numbers as counts*of*paths in a subgraph*of*the Cayley*graph**of*the symmetric group. ... For a pair*of*bold ends*of*weight µ*k*, assume without loss*of**generality*the two corresponding*cycles*are (1 . . . µ*k*)*and*(µ*k*+ 1 . . . 2µ*k*). ...##
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k -tuple chromatic number of the cartesian product of graphs

2015
*
Electronic Notes in Discrete Mathematics
*

A

doi:10.1016/j.endm.2015.07.041
fatcat:xwc7esxosrbdbpase2wvcd7otu
*k*-*tuple**coloring**of*a*graph*G assigns a set*of**k**colors*to each vertex*of*G such that if two vertices are adjacent, the corresponding sets*of**colors*are disjoint. ... In this paper, we show that there exist*graphs*G*and*H such that χ*k*(G H) > max{χ*k*(G), χ*k*(H)} for*k*≥ 2. ... It is well known that Kneser*graphs*are vertex transitive*graphs*[5] . In this paper, we show that equality (1) does not hold in*general*for*k*-*tuple**colorings**of**graphs*. ...##
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Constraints, MMSNP and expander relational structures
[article]

2013
*
arXiv
*
pre-print

The technical novelty

arXiv:0706.1701v2
fatcat:acmazpht3ze6ne6v6qkkxak4cm
*of*the paper is a concept*of*expander relations*and*a new type*of*product for relational structures: a*generalization**of*the zig-zag product, the twisted product. ... We give a poly-time construction for a combinatorial classic known as Sparse Incomparability Lemma, studied by Erdos, Lovasz, Nesetril, Rodl*and**others*: We show that every Constraint Satisfaction Problem ... The image*of**other**cycles*with length < max{*k*, 2j} is still a*cycle*,*and*its cutting pairs are the same. ...##
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Small Linear Dependencies for Binary Vectors of Low Weight
[chapter]

2008
*
Bolyai Society Mathematical Studies
*

Our proof is based on showing that in every

doi:10.1007/978-3-540-85221-6_9
fatcat:egmlv4ydwbdyhhscrohxk57wji
*graph**of*average degree at least c log log n, every legal edge*coloring*produces a*cycle*in which one*of*the*colors*appears either once or twice. ... We show that every set*of*m cn √ n log log n vectors in {0, 1} n in which every vector has Hamming weight 3 contains a subset*of*O(log n) vectors that form a linear dependency. ... Part*of*this work was done at Microsoft Research, Redmond, Washington. I thank Assaf Naor for bringing [11] to my attention,*and*suggesting that it is related to the work in [6] . ...##
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A tri-tuple coordinate system derived for fast and accurate analysis of the colored de Bruijn graph-based pangenomes

2021
*
BMC Bioinformatics
*

Results We developed a new method, a

doi:10.1186/s12859-021-04149-w
pmid:34044757
fatcat:p2ytujwj6zdvhfwjmvzd3qrmwe
*colored*superbubble (cSupB), that can overcome the complexity*of**graphs**and*organize a set*of*species- or population-specific haplotype sequences*of*interest. ... any*colored*directed acyclic*graph*. ... We also thank all members*of*the Lin lab for their valuable suggestions. ...##
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k-tuple colorings of the Cartesian product of graphs

2018
*
Discrete Applied Mathematics
*

A

doi:10.1016/j.dam.2017.02.003
fatcat:wfaxm5xtcbfabntsvjm6mmlxry
*k*-*tuple**coloring**of*a*graph*G assigns a set*of**k**colors*to each vertex*of*G such that if two vertices are adjacent, the corresponding sets*of**colors*are disjoint. ... In this paper, we show that there exist*graphs*G*and*H such that χ*k*(G H) > max{χ*k*(G), χ*k*(H)} for*k*≥ 2. ... In this paper, we show that the analogous*of*equality (1) for*k*-*tuple**colorings**of**graphs*does not hold in*general*. ...##
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On winning strategies in Ehrenfeucht-Fraïssé games

1997
*
Theoretical Computer Science
*

First, we give a simpler

doi:10.1016/s0304-3975(96)00015-1
fatcat:skofxqhmbjhydahj34otlgpcc4
*and*much easier-to-understand proof*of*Ajtai*and*Fagin's result that reachability in directed finite*graphs*is not in monadic NP. ... We present a powerful*and*versatile new sufficient condition for the second player (the "duplicator") to have a winning strategy in an Ehrenfeucht-Fra'isst game on*graphs*. ... Acknowledgements The authors are grateful to Joe Halpern, Phokion Kolaitis, Larry Stockmeyer,*and*Moshe Vardi for giving us useful comments on an earlier version*of*the paper. ...##
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The Optimality of Partial Clique Covering for Index Coding
[article]

2018
*
arXiv
*
pre-print

Partial clique covering is one

arXiv:1701.06181v2
fatcat:jjr2uyq5jfhj7gki6uj2e4r2eq
*of*the most basic coding schemes for index coding problems,*generalizing*clique*and**cycle*covering on the side information digraph*and*further reducing the achievable broadcast ... We further extend to the*general*partial clique covering, offering sufficient conditions*of*its optimality*and*sub-optimality with the aid*of*strong connectivity decomposition. ... [16] , (fractional) local*graph**coloring*[11] , [25] as well as the recent development*of*fractional local partial clique covering [26]*and*interlinked*cycle*covering [27] , among many*others*. ...
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