Square-free graphs are multiplicative

Wrochna, Marcin

doi:10.1016/j.jctb.2016.07.007

Cited by 11 publications

(20 citation statements)

References 27 publications

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“…Matsushita [Mat19] used the box complex to show that Hedetniemi's conjecture would imply an analogous conjecture in topology. This was independently proved by the first author [Wro19] using Ω k functors, while the box complex was used to show that squarefree graphs are multiplicative [Wro17]. See [FT18] for a survey on applications of adjoint functors to the conjecture.…”

Section: Hedetniemi's Conjecturementioning

confidence: 98%

Improved hardness for H-colourings of G-colourable graphs

Wrochna

Živný

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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We present new results on approximate colourings of graphs and, more generally, approximate H-colourings and promise constraint satisfaction problems.First, we show NP-hardness of colouring k-colourable graphs with k k/2 −1 colours for every k ≥ 4. This improves the result of Bulín, Krokhin, and Opršal [STOC'19], who gave NP-hardness of colouring k-colourable graphs with 2k−1 colours for k ≥ 3, and the result of Huang [APPROX-RANDOM'13], who gave NP-hardness of colouring k-colourable graphs with 2 Ω(k 1/3 ) colours for sufficiently large k. Thus, for k ≥ 4, we improve from known linear/subexponential gaps to exponential gaps.Second, we show that the topology of the box complex of H alone determines whether H-colouring of G-colourable graphs is NP-hard for all (non-bipartite, H-colourable) G. This formalises the topological intuition behind the result of Krokhin and Opršal [FOCS'19] that 3-colouring of G-colourable graphs is NP-hard for all (3-colourable, non-bipartite) G. We use this technique to establish NP-hardness of H-colouring of G-colourable graphs for H that include but go beyond K 3 , including square-free graphs and circular cliques (leaving K 4 and larger cliques open).Underlying all of our proofs is a very general observation that adjoint functors give reductions between promise constraint satisfaction problems.

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Section: Hedetniemi's Conjecturementioning

confidence: 98%

Improved hardness for H-colourings of G-colourable graphs

Wrochna

Živný

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In [Tar05], this result is used as a black box to find new multiplicative graphs from known ones, using adjoint functors. In [Wro17b], the use of the result is combined with an elaboration of the proof technique to prove that square-free graphs are strongly multiplicative. We review and modify this approach here, and emphasise the fact that it is strong multiplicativity being established.…”

Section: Multiplicativity and Strong Multiplicativitymentioning

confidence: 99%

“…As introduced in the previous section, an important part of proving that a graph K is strongly multiplicative is to show that the component K Cn ε of K Cn admits a homomorphism to K, for odd n. In fact, the topological approach from [Wro17b] shows that is is enough to prove this property, and to show strong multiplicativity for all unicyclic covers of K, which are often much simpler than K.…”

Section: Properties That Imply Strong Multiplicativitymentioning

confidence: 99%

“…The proof of Theorem 3 directly generalizes to many more graphs, as long as we consider walks up to a certain equivalence relation, see Theorem 10 later. We defer the proof, in the more general version, to Appendix A, as it only rephrases arguments of [Wro17b].…”

Section: Properties That Imply Strong Multiplicativitymentioning

confidence: 99%

“…Then in 2005, the first author [Tar05] proved that all circular cliques K r , r ∈ [2, 4) are multiplicative, and in 2017 the second author [Wro17b] proved that all square-free graphs are multiplicative. With this, it might look as though the techniques for handling multiplicativity are slowly expanding, and perhaps some day it will be possible to show that K 4 and the other complete graphs are multiplicative, proving Hedetniemi's conjecture.…”

Section: Introductionmentioning

confidence: 99%

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Hedetniemi's Conjecture and Strongly Multiplicative Graphs

Tardif¹,

Wrochna²

2019

SIAM J. Discrete Math.

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A graph K is multiplicative if a homomorphism from any product G × H to K implies a homomorphism from G or from H. Hedetniemi's conjecture states that all cliques are multiplicative. In an attempt to explore the boundaries of current methods, we investigate strongly multiplicative graphs, which we define as K such that for any connected graphs G, H with odd cycles C, C , a homomorphism fromStrong multiplicativity of K also implies the following property, which may be of independent interest: if G is non-bipartite, H is a connected graph with a vertex h, and there is a homomorphism φ : G × H → K such that φ(−, h) is constant, then H admits a homomorphism to K.All graphs currently known to be multiplicative are strongly multiplicative. We revisit the proofs in a different view based on covering graphs and replace fragments with more combinatorial arguments. This allows us to find new (strongly) multiplicative graphs: all graphs in which every edge is in at most square, and the third power of any graph of girth > 12. Though more graphs are amenable to our methods, they still make no progress for the case of cliques. Instead we hope to understand their limits, perhaps hinting at ways to further extend them.

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Graph homomorphism reconfiguration and frozen H‐colorings

Brewster

Lee

Moore

et al. 2019

Journal of Graph Theory

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For a fixed graph H, the reconfiguration problem for H-colourings (i.e. homomorphisms to H) asks: given a graph G and two H-colourings ϕ and ψ of G, does there exist a sequence f0, . . . , fm of H-colourings such that f0 = ϕ, fm = ψ and fi(u)fi+1(v) ∈ E(H) for every 0 ≤ i < m and uv ∈ E(G)? If the graph G is loop-free, then this is the equivalent to asking whether it possible to transform ϕ into ψ by changing the colour of one vertex at a time such that all intermediate mappings are H-colourings. In the affirmative, we say that ϕ reconfigures to ψ. Currently, the complexity of deciding whether an H-colouring ϕ reconfigures to an H-colouring ψ is only known when H is a clique, a circular clique, a C4-free graph, or in a few other cases which are easily derived from these. We show that this problem is PSPACE-complete when H is an odd wheel.An important notion in the study of reconfiguration problems for H-colourings is that of a frozen H-colouring; i.e. an H-colouring ϕ such that ϕ does not reconfigure to any Hcolouring ψ such that ψ = ϕ. We obtain an explicit dichotomy theorem for the problem of deciding whether a given graph G admits a frozen H-colouring. The hardness proof involves a reduction from a CSP problem which is shown to be NP-complete by establishing the non-existence of a certain type of polymorphism.

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Square-free graphs are multiplicative

Cited by 11 publications

References 27 publications

Improved hardness for H-colourings of G-colourable graphs

Improved hardness for H-colourings of G-colourable graphs

Hedetniemi's Conjecture and Strongly Multiplicative Graphs

Graph homomorphism reconfiguration and frozen H‐colorings

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