25,154 research outputs found
Brick polytopes, lattice quotients, and Hopf algebras
This paper is motivated by the interplay between the Tamari lattice, J.-L.
Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf
algebra on binary trees. We show that these constructions extend in the world
of acyclic -triangulations, which were already considered as the vertices of
V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural
surjection from the permutations to the acyclic -triangulations. We show
that the fibers of this surjection are the classes of the congruence
on defined as the transitive closure of the rewriting rule for letters
and words on . We then
show that the increasing flip order on -triangulations is the lattice
quotient of the weak order by this congruence. Moreover, we use this surjection
to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on
permutations, indexed by acyclic -triangulations, and to describe the
product and coproduct in this algebra and its dual in term of combinatorial
operations on acyclic -triangulations. Finally, we extend our results in
three directions, describing a Cambrian, a tuple, and a Schr\"oder version of
these constructions.Comment: 59 pages, 32 figure
Brick assignments and homogeneously almost self-complementary graphs
AbstractA graph is called almost self-complementary if it is isomorphic to the graph obtained from its complement by removing a 1-factor. In this paper, we study a special class of vertex-transitive almost self-complementary graphs called homogeneously almost self-complementary. These graphs occur as factors of symmetric index-2 homogeneous factorizations of the “cocktail party graphs” K2n−nK2. We construct several infinite families of homogeneously almost self-complementary graphs, study their structure, and prove several classification results, including the characterization of all integers n of the form n=pr and n=2p with p prime for which there exists a homogeneously almost self-complementary graph on 2n vertices
Sharing Stories as Legacy: What Matters to Older Adults?
Objectives: Legacy allows individuals to make meaning of their lives by passing on their experiences and beliefs to younger people and influencing their perspectives, perceptions, and actions. This mixed-methods study investigated: (1) What is important for older adults to share as legacy with families, friends and others, based on the types and features of their digital stories ? and (2) How do older adults’ digital stories affect story viewers?
Methods: One hundred adults aged between 55 and 95 years participated in ten-week Elder’s Digital Storytelling courses and created short digital stories. Using the content analysis approach, the story transcripts were thematically analyzed and iteratively coded by three researchers and the results were quantified. A diverse group of 60 viewers at a public event provided their reactions to the digital stories.
Results: The findings revealed that character, place, and family were chosen as the primary types by the older adults for their legacy digital stories. Accomplishment and career/school were the next most prominent story types. Moreover, these digital stories appeared to have a powerful impact on the viewers.
Discussion: A digital story is a powerful artifact to communicate an older person’s legacy because it is based on familiar forms of communication, such as speech and photographs. The major legacy themes chosen by the older adults align with the findings of the research literature. The feedback from the viewers of the digital stories reflects these as a source of life wisdom and legacy for younger generations.
Funding details: This work was supported by the AGE-WELL National Centre of Excellence (AW CRP 2015-WP4.3)
The brick polytope of a sorting network
The associahedron is a polytope whose graph is the graph of flips on
triangulations of a convex polygon. Pseudotriangulations and
multitriangulations generalize triangulations in two different ways, which have
been unified by Pilaud and Pocchiola in their study of flip graphs on
pseudoline arrangements with contacts supported by a given sorting network.
In this paper, we construct the brick polytope of a sorting network, obtained
as the convex hull of the brick vectors associated to each pseudoline
arrangement supported by the network. We combinatorially characterize the
vertices of this polytope, describe its faces, and decompose it as a Minkowski
sum of matroid polytopes.
Our brick polytopes include Hohlweg and Lange's many realizations of the
associahedron, which arise as brick polytopes for certain well-chosen sorting
networks. We furthermore discuss the brick polytopes of sorting networks
supporting pseudoline arrangements which correspond to multitriangulations of
convex polygons: our polytopes only realize subgraphs of the flip graphs on
multitriangulations and they cannot appear as projections of a hypothetical
multiassociahedron.Comment: 36 pages, 25 figures; Version 2 refers to the recent generalization
of our results to spherical subword complexes on finite Coxeter groups
(http://arxiv.org/abs/1111.3349
A structure theorem for product sets in extra special groups
Hegyv\'ari and Hennecart showed that if is a sufficiently large brick of
a Heisenberg group, then the product set contains many cosets of the
center of the group. We give a new, robust proof of this theorem that extends
to all extra special groups as well as to a large family of quasigroups.Comment: This manuscript has been updated to include referee corrections. To
appear in Journal of Number Theor
Rainbow Connection Number in the Brick Product Graphs C(2n,m, r)
All graphs considered in this paper are simple, finite and undirected. Let G be a nontrivial connected graph with an edge coloring c : E(G) → {1, 2, · · · , k}, k ∈ N, where adjacent edges may be colored the same. A path in G is called a rainbow path if no two edges of it are colored the same
Snarks with total chromatic number 5
A k-total-coloring of G is an assignment of k colors to the edges and vertices of G, so that adjacent and incident elements have different colors. The total chromatic number of G, denoted by chi(T)(G), is the least k for which G has a k-total-coloring. It was proved by Rosenfeld that the total chromatic number of a cubic graph is either 4 or 5. Cubic graphs with chi(T) = 4 are said to be Type 1, and cubic graphs with chi(T) = 5 are said to be Type 2.
Snarks are cyclically 4-edge-connected cubic graphs that do not allow a 3-edge-coloring. In 2003, Cavicchioli et al. asked for a Type 2 snark with girth at least 5. As neither Type 2 cubic graphs with girth at least 5 nor Type 2 snarks are known, this is taking two steps at once, and the two requirements of being a snark and having girth at least 5 should better be treated independently.
In this paper we will show that the property of being a snark can be combined with being Type 2. We will give a construction that gives Type 2 snarks for each even vertex number n >= 40.
We will also give the result of a computer search showing that among all Type 2 cubic graphs on up to 32 vertices, all but three contain an induced chordless cycle of length 4. These three exceptions contain triangles. The question of the existence of a Type 2 cubic graph with girth at least 5 remains open
- …