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Intertwining numbers; the n -rowed shapes

Hyoung J. Ko, Kyoung J. Lee (2007)

Czechoslovak Mathematical Journal

A fairly old problem in modular representation theory is to determine the vanishing behavior of the H o m groups and higher E x t groups of Weyl modules and to compute the dimension of the / ( p ) -vector space H o m A ¯ r ( K ¯ λ , K ¯ μ ) for any partitions λ , μ of r , which is the intertwining number. K. Akin, D. A. Buchsbaum, and D. Flores solved this problem in the cases of partitions of length two and three. In this paper, we describe the vanishing behavior of the groups H o m A ¯ r ( K ¯ λ , K ¯ μ ) and provide a new formula for the intertwining number for any...

Interval edge colorings of some products of graphs

Petros A. Petrosyan (2011)

Discussiones Mathematicae Graph Theory

An edge coloring of a graph G with colors 1,2,...,t is called an interval t-coloring if for each i ∈ {1,2,...,t} there is at least one edge of G colored by i, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. A graph G is interval colorable, if there is an integer t ≥ 1 for which G has an interval t-coloring. Let ℜ be the set of all interval colorable graphs. In 2004 Kubale and Giaro showed that if G,H ∈ 𝔑, then the Cartesian product of these graphs...

Interval Edge-Colorings of Cartesian Products of Graphs I

Petros A. Petrosyan, Hrant H. Khachatrian, Hovhannes G. Tananyan (2013)

Discussiones Mathematicae Graph Theory

A proper edge-coloring of a graph G with colors 1, . . . , t is an interval t-coloring if all colors are used and the colors of edges incident to each vertex of G form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. Let [...] be the set of all interval colorable graphs. For a graph G ∈ [...] , the least and the greatest values of t for which G has an interval t-coloring are denoted by w(G) and W(G), respectively. In this paper...

Interval Incidence Coloring of Subcubic Graphs

Anna Małafiejska, Michał Małafiejski (2017)

Discussiones Mathematicae Graph Theory

In this paper we study the problem of interval incidence coloring of subcubic graphs. In [14] the authors proved that the interval incidence 4-coloring problem is polynomially solvable and the interval incidence 5-coloring problem is NP-complete, and they asked if Xii(G) ≤ 2Δ(G) holds for an arbitrary graph G. In this paper, we prove that an interval incidence 6-coloring always exists for any subcubic graph G with Δ(G) = 3.

Intrinsic linking and knotting are arbitrarily complex

Erica Flapan, Blake Mellor, Ramin Naimi (2008)

Fundamenta Mathematicae

We show that, given any n and α, any embedding of any sufficiently large complete graph in ℝ³ contains an oriented link with components Q₁, ..., Qₙ such that for every i ≠ j, | l k ( Q i , Q j ) | α and | a ( Q i ) | α , where a ( Q i ) denotes the second coefficient of the Conway polynomial of Q i .

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