$B$-valuations of graphs

Havel, Ivan M.; Morávek, Jaroslav

doi:10.21136/cmj.1972.101102

Cited by 56 publications

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“…In the above definitions, we often leave out the explicit reference to a coloring ϕ, if the coloring is clear from the context. [20] (see also [19]) proved a criterion for a graph G to be a subgraph of a hypercube: Proposition 2.1. A graph G is a subgraph of Q d if and only if there is a proper d-edge coloring of G with integers {1, .…”

Section: Preliminariesmentioning

confidence: 99%

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Restricted extension of sparse partial edge colorings of hypercubes

Casselgren

Markström

Pham

2020

Discrete Mathematics

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We consider the problem of extending and avoiding partial edge colorings of hypercubes; that is, given a partial edge coloring ϕ of the d-dimensional hypercube Q d , we are interested in whether there is a proper d-edge coloring of Q d that agrees with the coloring ϕ on every edge that is colored under ϕ; or, similarly, if there is a proper d-edge coloring that disagrees with ϕ on every edge that is colored under ϕ. In particular, we prove that for any d ≥ 1, if ϕ is a partial d-edge coloring of Q d , then ϕ is avoidable if every color appears on at most d/8 edges and the coloring satisfies a relatively mild structural condition, or ϕ is proper and every color appears on at most d − 2 edges. We also show that the same conclusion holds if d is divisible by 3 and every color class of ϕ is an induced matching. Moreover, for all 1 ≤ k ≤ d, we characterize for which configurations consisting of a partial coloring ϕ of d − k edges and a partial coloring ψ of k edges, there is an extension of ϕ that avoids ψ.

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Section: Preliminariesmentioning

confidence: 99%

“…Havel and Moravek [20] (see also [19]) proved a criterion for a graph G to be a subgraph of a hypercube: Proposition 2.1. A graph G is a subgraph of Q d if and only if there is a proper d-edge coloring of G with integers {1, .…”

Section: Preliminariesmentioning

confidence: 99%

Restricted extension of sparse partial edge colorings of hypercubes

Casselgren

Markström

Pham

2020

Discrete Mathematics

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“…Havel and Moravek [13] (see also [12]) proved a criterion for a graph

G

to be a subgraph of a hypercube:…”

Section: Preliminariesmentioning

confidence: 99%

Edge precoloring extension of hypercubes

Casselgren

Markström

Pham

2020

Journal of Graph Theory

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We consider the problem of extending partial edge colorings of hypercubes. In particular, we obtain an analogue of the positive solution to the famous Evans' conjecture on completing partial Latin squares by proving that every proper partial edge coloring of at most d − 1 edges of the d-dimensional hypercube Q d can be extended to a proper d-edge coloring of Q d . Additionally, we characterize which partial edge colorings of Q d with precisely d precolored edges are extendable to proper d-edge colorings of Q d , and consider some related edge precoloring extension problems of hypercubes.Note that the conjecture on distance-2 matchings in [6] is sharp both with respect to the distance between precolored edges, and in the sense that ∆(G) + 1 can in general not be replaced by ∆(G), even if any two precolored edges are at arbitrarily large distance from each other [6]. In [6], it is proved that this conjecture holds for e.g. bipartite multigraphs and subcubic multigraphs, and in [11] it is proved that a version of the conjecture with the distance increased to 9 holds for general graphs.However, for one specific family of graphs, the balanced complete bipartite graphs K n,n , the edge precoloring extension problem was studied far earlier than in the above-mentioned references.Here the extension problem corresponds to asking whether a partial Latin square can be completed to a Latin square. In this form the problem appeared already in 1960, when Evans [7] stated his now classic conjecture that for every positive integer n, if n−1 edges in K n,n have been (properly) colored, then the partial coloring can be extended to a proper n-edge-coloring of K n,n . This conjecture was solved for large n by Häggkvist [15] and later for all n by Smetaniuk [18], and independently by Andersen and Hilton [2]. Moreover, Andersen and Hilton [2] characterized which n × n partial Latin squares with exactly n non-empty cells are extendable.In this paper we consider the edge precoloring extension problem for the family of hypercubes. Although matching extendability and subgraph containment problems have been studied extensively for hypercubes, (see e.g. [19,9,20] and references therein) the edge precoloring extension problem for hypercubes seems to be a hitherto quite unexplored line of research. As in the setting of completing partial Latin squares (and unlike the papers [6, 11]) we consider only proper edge colorings of hypercubes Q d using exactly ∆(Q d ) colors.We prove that every proper edge precoloring of the d-dimensional hypercube Q d with at most d − 1 precolored edges is extendable to a d-edge coloring of Q d , thereby establishing an analogue of the positive resolution of Evans' conjecture. Moreover, similarly to [2] we also characterize which proper precolorings with exactly d precolored edges are not extendable to proper d-edge colorings of Q d . We also consider the cases when the precolored edges form an induced matching or one or two hypercubes of smaller dimension. The paper is concluded by a conjecture and some examples and...

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“…Havel and Moravek [29] found a criterion for a graph G to be cubical based on so-called c-valuations. A c-valuation of a bipartite graph G is a labeling of the edges of G such that • for each cycle in G, all distinct edge labels occur an even number of times; • for each (noncyclic) path in G, there exists at least one edge label which occurs an odd number of times.…”

Section: -Connectednessmentioning

confidence: 99%

“…The dimension of a c-valuation is the number of edge labels used. It is shown in [29] that a graph G is cubical with G ⊆ Q n if and only if there exists a c-valuation of G of dimension n. Intuitively, the labels of the edges correspond with the directions of the edges in an n-cube embedding of G.…”

Section: -Connectednessmentioning

confidence: 99%

Representation of finite graphs as difference graphs of S-units, I

Győry

Hajdu

Tijdeman

2014

Journal of Combinatorial Theory, Series A

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Abstract. Let S be a finite non-empty set of primes, Z S the ring of those rationals whose denominators are not divisible by primes outside S, and Z * S the multiplicative group of invertible elements (S-units) in Z S . For a non-empty subset A of Z S , denote by G S (A) the graph with vertex set A and with an edge between a and b if and only if a − b ∈ Z * S . This type of graphs has been studied by many people.In the present paper we deal with the representability of finite (simple) graphs G as G S (A). If A = uA + a for some u ∈ Z * S and a ∈ Z S , then A and A are called S-equivalent, since G S (A) and G S (A ) are isomorphic. We say that a finite graph G is representable / infinitely representable with S if G is isomorphic to G S (A) for some A / for infinitely many non-S-equivalent A.We prove among other things that for any finite graph G there exist infinitely many finite sets S of primes such that G can be represented with S. We deal with the infinite representability of finite graphs, in particular cycles and complete bipartite graphs. Further, we consider the triangles in G for a deeper analysis. Finally, we prove that G is representable with every S if and only if G is cubical.Besides combinatorial and numbertheoretical arguments, some deep Diophantine results concerning S-unit equations are used in our proofs.In Part II, we shall investigate these and similar problems over more general domains.2010 Mathematics Subject Classification. 05C25, 05C62, 11D61.

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$B$-valuations of graphs

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Cited by 56 publications

References 1 publication

Restricted extension of sparse partial edge colorings of hypercubes

Restricted extension of sparse partial edge colorings of hypercubes

Edge precoloring extension of hypercubes

Representation of finite graphs as difference graphs of S-units, I

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