On topological relaxations of chromatic conjectures

Simonyi, Gábor; Zsbán, Ambrus

doi:10.1016/j.ejc.2010.06.001

Cited by 16 publications

(14 citation statements)

References 34 publications

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“…♦ Remark 3. A more sophisticated such variable as the ones mentioned in Remark 2 is a topological relaxation of the chromatic number in the sense of [45]. There are several closely related lower bounds on the chromatic number based on algebraic topology that all grew out from the pioneering work of Lovász in [32].…”

Section: On the Possibilities Of Equalitymentioning

confidence: 99%

“…There are several closely related lower bounds on the chromatic number based on algebraic topology that all grew out from the pioneering work of Lovász in [32]. In [45] the topological lower bound involving the so-called co-index of the box complex of the graph at hand is considered as a graph parameter itself and it is shown that it also satisfies the Hedetniemi-type equality.…”

Section: On the Possibilities Of Equalitymentioning

confidence: 99%

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Shannon Capacity and the Categorical Product

Simonyi

2021

Electron. J. Combin.

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Shannon OR-capacity $C_{\rm OR}(G)$ of a graph $G$, that is the traditionally more often used Shannon AND-capacity of the complementary graph, is a homomorphism monotone graph parameter therefore $C_{\rm OR}(F\times G)\leqslant\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$ holds for every pair of graphs, where $F\times G$ is the categorical product of graphs $F$ and $G$. Here we initiate the study of the question when could we expect equality in this inequality. Using a strong recent result of Zuiddam, we show that if this "Hedetniemi-type" equality is not satisfied for some pair of graphs then the analogous equality is also not satisfied for this graph pair by some other graph invariant that has a much "nicer" behavior concerning some different graph operations. In particular, unlike Shannon OR-capacity or the chromatic number, this other invariant is both multiplicative under the OR-product and additive under the join operation, while it is also nondecreasing along graph homomorphisms. We also present a natural lower bound on $C_{\rm OR}(F\times G)$ and elaborate on the question of how to find graph pairs for which it is known to be strictly less than the upper bound $\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$. We present such graph pairs using the properties of Paley graphs.

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Section: On the Possibilities Of Equalitymentioning

confidence: 99%

Section: On the Possibilities Of Equalitymentioning

confidence: 99%

Shannon Capacity and the Categorical Product

Simonyi

2021

Electron. J. Combin.

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“…The dual to Conjecture 1.6, namely coind(X × Y ) = min(coind(X), coind(Y )), has been considered by Simonyi and Zsbán [SZ10]. This statement is trivial in topology, that is, S n → Z 2 X × Y if and only if S n → Z 2 X and S n → Z 2 Y .…”

Section: Obstacles and Non-tidy Spacesmentioning

confidence: 99%

On inverse powers of graphs and topological implications of Hedetniemi's conjecture

Wrochna

2019

Journal of Combinatorial Theory, Series B

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We consider a natural graph operation Ω k that is a certain inverse (formally: the right adjoint) to taking the k-th power of a graph. We show that it preserves the topology (the Z 2 -homotopy type) of the box complex, a basic tool in applications of topology in combinatorics. Moreover, we prove that the box complex of a graph G admits a Z 2 -map (an equivariant, continuous map) to the box complex of a graph H if and only if the graph Ω k (G) admits a homomorphism to H, for high enough k.This allows to show that if Hedetniemi's conjecture on the chromatic number of graph products were true for n-colorings, then the following analogous conjecture in topology would also be true: If X, Y are Z 2 -spaces (finite Z 2 -simplicial complexes) such that X × Y admits a Z 2 -map to the (n − 2)-dimensional sphere, then X or Y itself admits such a map. We discuss this and other implications, arguing the importance of the topological conjecture. arXiv:1712.03196v2 [math.CO]

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“…Hell [4] proved that if the topological bound is tight for all factors in a product of c-chromatic graphs, then the product is c-chromatic. (See also [10] for a contemporary treatment of the subject.) Problem 4.2.…”

Section: Proofmentioning

confidence: 99%

The chromatic Ramsey number of odd wheels

Paul

Tardif

2011

Journal of Graph Theory

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Abstract. We prove that the chromatic Ramsey number of every odd wheel W 2k+1 , k ≥ 2 is 14. That is, for every odd wheel W 2k+1 , there exists a 14-chromatic graph F such that when the edges of F are two-coloured, there is a monochromatic copy of W 2k+1 in F , and no graph F with chromatic number 13 has the same property. We ask whether a natural extension of odd wheels to the family of generalized Mycielski graphs could help to prove the Burr-Erdős-Lovász on the minimum possible chromatic Ramsey number of a n-chromatic graph.

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On topological relaxations of chromatic conjectures

Cited by 16 publications

References 34 publications

Shannon Capacity and the Categorical Product

Shannon Capacity and the Categorical Product

On inverse powers of graphs and topological implications of Hedetniemi's conjecture

The chromatic Ramsey number of odd wheels

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