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### Generalized k-tuple colorings of cycles and other graphs

R.C Brigham, R.D Dutton
1982 Journal of combinatorial theory. Series B (Print)
Stahl's concept 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  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.  ...

### n-Tuple colorings and associated graphs

Saul Stahl
1976 Journal of combinatorial theory. Series B (Print)
In G, Figure 4 contains 1 -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

Jean-Luc Baril, Olivier Togni
2009 Discrete Mathematics
This paper studies proper 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.  ...

### Packing Loose Hamilton Cycles [article]

Asaf Ferber, Kyle Luh, Daniel Montealegre, Oanh Nguyen
2016 arXiv   pre-print
A subset C 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.  ...

### On k-tuple and k-tuple total domination numbers of regular graphs [article]

Sharareh Alipour, Amir Jafari, Morteza Saghafian
2018 arXiv   pre-print
Henning 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  , 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  ...

### On the Combinatorial Power of the Weisfeiler-Lehman Algorithm [article]

Martin Fürer
2017 arXiv   pre-print
We focus on two fundamental invariants, the num- ber of cycles Cp of length p, and the number of cliques Kp of size p.  ...  The classical Weisfeiler-Lehman method WL 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.  ...

### Multicoloring and Mycielski construction

Wensong Lin
2008 Discrete Mathematics
We then investigate the kth chromatic number 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  ...

### The Combinatorics of Real Double Hurwitz Numbers with Real Positive Branch Points

Mathieu Guay-Paquet, Hannah Markwig, Johannes Rau
2015 International mathematics research notices
us to uncover the beautiful combinatorics 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 ) andk + 1 . . . 2µ k ).  ...

### k -tuple chromatic number of the cartesian product of graphs

Flavia Bonomo, Ivo Koch, Pablo Torres, Mario Valencia-Pabon
2015 Electronic Notes in Discrete Mathematics
A 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  . In this paper, we show that equality (1) does not hold in general for k-tuple colorings of graphs.  ...

### Constraints, MMSNP and expander relational structures [article]

Gabor Kun
2013 arXiv   pre-print
The technical novelty 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.  ...

### Small Linear Dependencies for Binary Vectors of Low Weight [chapter]

Uriel Feige
2008 Bolyai Society Mathematical Studies
Our proof is based on showing that in every 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  to my attention, and suggesting that it is related to the work in  .  ...

### A tri-tuple coordinate system derived for fast and accurate analysis of the colored de Bruijn graph-based pangenomes

Jindan Guo, Erli Pang, Hongtao Song, Kui Lin
2021 BMC Bioinformatics
Results We developed a new method, a 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.  ...

### k-tuple colorings of the Cartesian product of graphs

Flavia Bonomo, Ivo Koch, Pablo Torres, Mario Valencia-Pabon
2018 Discrete Applied Mathematics
A 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.  ...

### On winning strategies in Ehrenfeucht-Fraïssé games

Sanjeev Arora, Ronald Fagin
1997 Theoretical Computer Science
First, we give a simpler 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.  ...

### The Optimality of Partial Clique Covering for Index Coding [article]

Xinping Yi, Giuseppe Caire
2018 arXiv   pre-print
Partial clique covering is one 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.  ...   , (fractional) local graph coloring  ,  as well as the recent development of fractional local partial clique covering  and interlinked cycle covering  , among many others.  ...
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