The Chromatic Index of Projective Triple Systems

Meszka, Mariusz

doi:10.1002/jcd.21368

Cited by 7 publications

(14 citation statements)

References 8 publications

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“…Finally, one may investigate various combinatorial properties of 6-valent first-kind Frobenius circulants and EJ graphs in general. The results in [28] imply that the chromatic number of any 6-valent first-kind Frobenius circulant Γ of order n ≡ 1 mod 6 is equal to 4 if n = 7, 13, 19, and 7, 5, 5 respectively in these exceptional cases. Thus, if n > 19, then the independence number of Γ is at least n/4 .…”

Section: Discussionmentioning

confidence: 92%

Frobenius circulant graphs of valency six, Eisenstein–Jacobi networks, and hexagonal meshes

Thomson

Zhou

2014

European Journal of Combinatorics

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A Frobenius group is a transitive but not regular permutation group such that only the identity element can fix two points. A finite Frobenius group can be expressed as G = K H with K a nilpotent normal subgroup. A first-kind G-Frobenius graph is a Cayley graph on K with connection set S an H-orbit on K generating K, where H is of even order or S consists of involutions. We classify all 6-valent first-kind Frobenius circulant graphs such that the underlying kernel K is cyclic. We give optimal gossiping and routing algorithms for such a circulant and compute its forwarding indices, Wiener indices and minimum gossip time. We also prove that its broadcasting time is equal to its diameter plus two or three. We prove that all 6-valent first-kind Frobenius circulants with cyclic kernels are Eisenstein-Jacobi graphs, the latter being Cayley graphs on quotient rings of the ring of Eisenstein-Jacobi integers. We also prove that larger Eisenstein-Jacobi graphs can be constructed from smaller ones as topological covers, and a similar result holds for 6-valent first-kind Frobenius circulants. As a corollary any Eisenstein-Jacobi graph with order congruent to 1 modulo 6 and underlying Eisenstein-Jacobi integer not an associate of a real integer, is a cover of a 6-valent first-kind Frobenius circulant. A distributed real-time computing architecture known as HARTS or hexagonal mesh is a special 6-valent first-kind Frobenius circulant.

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Section: Discussionmentioning

confidence: 92%

Frobenius circulant graphs of valency six, Eisenstein–Jacobi networks, and hexagonal meshes

Thomson

Zhou

2014

European Journal of Combinatorics

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“…This paper generalizes Meszka's work in [17] to establish a general framework to count the exact value of χ ′ (PG(n, q)) for any integer n 5 and any prime power q. Two properties of PG(n, q) are discussed in Section 2.…”

Section: Introductionmentioning

confidence: 84%

“…Meszka [17,Lemma 2] pointed out that PG(3, 2) has a special property, which can be used to determine the exact value of χ ′ (PG(n, 2)) for any n > 3. In this section, we generalize this property to PG(3, q) for any prime power q.…”

Section: Property Ementioning

confidence: 99%

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The chromatic index of finite projective spaces

Xu¹,

Feng²

2022

Preprint

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A coloring of PG(n, q), the n-dimensional projective geometry over GF(q), is an assignment of colors to all lines of PG(n, q) so that any two lines with the same color do not intersect. The chromatic index of PG(n, q), denoted by χ ′ (PG(n, q)), is the least number of colors for which a coloring of PG(n, q) exists. This paper translates the problem of determining the chromatic index of PG(n, q) to the problem of examining the existences of PG(3, q) and PG(4, q) with certain properties. As applications, it is shown that for any odd integer n and q ∈ {3, 4, 8}, χ ′ (PG(n, q)) = n 1 q , which implies the existence of a parallelism of PG(n, q) for any odd integer n and q ∈ {3, 4, 8}.

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“…Two sparse infinite families of such Steiner triple systems have been found, one by Wilson (see [9]) and one by the authors [2]. Also see [6] for a recent related result.…”

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confidence: 99%

Steiner Triple Systems without Parallel Classes

Bryant

Horsley

2015

SIAM J. Discrete Math.

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We construct Steiner triple systems without parallel classes for an infinite number of orders congruent to 3 (mod 6). The only previously known examples have order 15 or 21.A Steiner triple system of order v is a pair (V, B) where V is a v-set of points and B is a collection of 3-subsets of V , called triples, such that each (unordered) pair of points occurs in exactly one triple.Steiner triple systems are fundamental in design theory. Kirkman [4] proved in 1847 that there exists a Steiner triple system of order v if and only if v ≡ 1 or 3 (mod 6) (see [3]).A parallel class in a Steiner triple system (V, B) is a subset of B that partitions V . Of course, only Steiner triple systems of order congruent to 3 (mod 6) may contain parallel classes. It is a longstanding conjecture, dating back to at least 1984, that systems without parallel classes exist for all orders congruent to 3 (mod 6) except 3 and 9 (see [3]). In spite of this, the only previously known examples without parallel classes have order 15 or 21. The unique Steiner triple system of order 9 has a parallel class, 10 of the 80 nonisomorphic Steiner triple systems of order 15 have no parallel classes (see [3]), and 12 of the 1772 nonisomorphic 4-rotational Steiner triple systems of order 21 have no parallel classes [5].In this paper we construct Steiner triple systems without parallel classes for an infinite number of orders congruent to 3 (mod 6). Our construction is similar to one used by Schreiber [10] and Wilson [11]. Problems concerning parallel classes in Steiner triple systems are well studied. Ray-Chaudhuri and Wilson [8] famously proved that for every order congruent to 3 (mod 6) there exists a Steiner triple system whose block set can be partitioned into parallel classes. The study of parallel classes in Steiner triples also relates to problems concerning matchings in certain hypergraphs, and Alon et al. [1] give a lower bound on the number of disjoint triples in any Steiner triple system. Also see [7] and the surveys [3] and [9].

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The Chromatic Index of Projective Triple Systems

Cited by 7 publications

References 8 publications

Frobenius circulant graphs of valency six, Eisenstein–Jacobi networks, and hexagonal meshes

Frobenius circulant graphs of valency six, Eisenstein–Jacobi networks, and hexagonal meshes

The chromatic index of finite projective spaces

Steiner Triple Systems without Parallel Classes

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