Graph Powers and Graph Homomorphisms

Hajiabolhassan, Hossein; Taherkhani, Ali

doi:10.37236/289

Cited by 21 publications

(38 citation statements)

References 28 publications

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“…More generally, for any odd cycle C k with k ≥ m n , we have L m n (C 2k+1 ) ↔ R m n (C 2k+1 ) ↔ K s/r , where s = nk and r = nk−m 2 . As noted in [13], the circular chromatic number of G can therefore be expressed in terms of the infimum of the values m n such that G → R m n (K 3 ). Since R m n is the thin right adjoint of L n m , we can look instead at the values m n such that L n m (G) admits a homomorphism to K 3 .…”

Section: Compositions and Chains Of Functorsmentioning

confidence: 99%

Hedetniemi’s Conjecture and Adjoint Functors in Thin Categories

Foniok

Tardif

2017

Appl Categor Struct

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Section: Compositions and Chains Of Functorsmentioning

confidence: 99%

Hedetniemi’s Conjecture and Adjoint Functors in Thin Categories

Foniok

Tardif

2017

Appl Categor Struct

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“…In this section we will prove Theorems 3 and 4. To start, let us recall the following results from [10] and [11].…”

Section: Proof (Of Theorem 2)mentioning

confidence: 99%

“…The rth power of a graph G, denoted by G r , is a graph on the vertex set V (G), in which two vertices are connected by an edge if there exist an r-walk between them in G. Also, the fractional power of a graph is defined as G Note that for simple graphs, G −→ H implies that G r −→ H r for any positive integer r > 0. Hence, Problem 1 is closely related to the study of the chromatic number of the third power of sparse 3-regular simple graphs (see [5,10,11]). The Petersen graph is an icon in graph theory.…”

Section: Introductionmentioning

confidence: 99%

On the odd girth and the circular chromatic number of generalized Petersen graphs

Daneshgar

Madani

2016

J Comb Optim

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A class G of simple graphs is said to be girth-closed (odd-girth-closed) if for any positive integer g there exists a graph G ∈ G such that the girth (odd-girth) of G is ≥ g. A girth-closed (odd-girth-closed) class G of graphs is said to be pentagonal (oddpentagonal) if there exists a positive integer g * depending on G such that any graph G ∈ G whose girth (odd-girth) is greater than g * admits a homomorphism to the five cycle (i.e. is C 5 -colourable). Although, the question "Is the class of simple 3-regular graphs pentagonal?" proposed by Nešetřil (Taiwan J Math 3:381-423, 1999) is still a central open problem, Gebleh (Theorems and computations in circular colourings of graphs, 2007) has shown that there exists an odd-girth-closed subclass of simple 3-regular graphs which is not odd-pentagonal. In this article, motivated by the conjecture that the class of generalized Petersen graphs is odd-pentagonal, we show that finding the odd girth of generalized Petersen graphs can be transformed to an integer programming problem, and using the combinatorial and number theoretic properties of this problem, we explicitly compute the odd girth of such graphs, showing that the class is odd-girthclosed. Also, we obtain upper and lower bounds for the circular chromatic number of these graphs, and as a consequence, we show that the subclass containing generalized Petersen graphs Pet(n, k) for which either k is even, n is odd and n k−1 ≡ ±2 or both n and k are odd and n ≥ 5k is odd-pentagonal. This in particular shows the existence of nontrivial odd-pentagonal subclasses of 3-regular simple graphs.A primary version of this article has already been posted on http://arxiv.org/abs/1501.06551.

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“…. , 5}, {(0, 1), (1,2), (3,2), (3,4), (4, 5)} , Λ T (T) is not homomorphically equivalent to a tree. Thus by Theorem 2.5, Γ T does not have a right adjoint.…”

Section: Example: the Arc Graph Constructionmentioning

confidence: 99%

Digraph functors which admit both left and right adjoints

Foniok

Tardif

2015

Discrete Mathematics

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For our purposes, two functors {\Lambda} and {\Gamma} are said to be respectively left and right adjoints of each other if for any digraphs G and H, there exists a homomorphism of {\Lambda}(G) to H if and only if there exists a homomorphism of G to {\Gamma}(H). We investigate the right adjoints characterised by Pultr in [A. Pultr, The right adjoints into the categories of relational systems, in Reports of the Midwest Category Seminar, IV, volume 137 of Lecture Notes in Mathematics, pages 100-113, Berlin, 1970]. We find necessary conditions for these functors to admit right adjoints themselves. We give many examples where these necessary conditions are satisfied, and the right adjoint indeed exists. Finally, we discuss a connection between these right adjoints and homomorphism dualities.Comment: 16 pages, 1 figur

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Graph Powers and Graph Homomorphisms

Cited by 21 publications

References 28 publications

Hedetniemi’s Conjecture and Adjoint Functors in Thin Categories

Hedetniemi’s Conjecture and Adjoint Functors in Thin Categories

On the odd girth and the circular chromatic number of generalized Petersen graphs

Digraph functors which admit both left and right adjoints

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