4-core partitions and class numbers
4-critical 4-valent planar graphs constructed with crowns.
4-cycle properties for characterizing rectagraphs and hypercubes
A -graph is a connected graph, where each pair of vertices has either 0 or 2 common neighbours. These graphs constitute a subclass of -graphs introduced by Mulder in 1979. A rectagraph, well known in diagram geometry, is a triangle-free -graph. -graphs include hypercubes, folded cube graphs and some particular graphs such as icosahedral graph, Shrikhande graph, Klein graph, Gewirtz graph, etc. In this paper, we give some local properties of 4-cycles in -graphs and more specifically in -graphs,...
4-Transitive Digraphs I: The Structure of Strong 4-Transitive Digraphs
Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A digraph D is transitive if for every three distinct vertices u, v,w ∈ V (D), (u, v), (v,w) ∈ A(D) implies that (u,w) ∈ A(D). This concept can be generalized as follows: A digraph is k-transitive if for every u, v ∈ V (D), the existence of a uv-directed path of length k in D implies that (u, v) ∈ A(D). A very useful structural characterization of transitive digraphs has been known for a long time, and...
5-sparse Steiner triple systems of order exist for almost all admissible .
5-Stars of Low Weight in Normal Plane Maps with Minimum Degree 5
It is known that there are normal plane maps M5 with minimum degree 5 such that the minimum degree-sum w(S5) of 5-stars at 5-vertices is arbitrarily large. In 1940, Lebesgue showed that if an M5 has no 4-stars of cyclic type (5, 6, 6, 5) centered at 5-vertices, then w(S5) ≤ 68. We improve this bound of 68 to 55 and give a construction of a (5, 6, 6, 5)-free M5 with w(S5) = 48
A 2-coloring of can have monochromatic Schur triples, but not less.
-analogue of some binomial coefficient identities of Y. Sun.
A backward selection procedure for approximating a discrete probability distribution by decomposable models
Decomposable (probabilistic) models are log-linear models generated by acyclic hypergraphs, and a number of nice properties enjoyed by them are known. In many applications the following selection problem naturally arises: given a probability distribution over a finite set of discrete variables and a positive integer , find a decomposable model with tree-width that best fits . If is the generating hypergraph of a decomposable model and is the estimate of under the model, we can measure...
A bi-average tree solution for probabilistic communication situations with fuzzy coalition
A probabilistic communication structure considers the setting with communication restrictions in which each pair of players has a probability to communicate directly. In this paper, we consider a more general framework, called a probabilistic communication structure with fuzzy coalition, that allows any player to have a participation degree to cooperate within a coalition. A maximal product spanning tree, indicating a way of the greatest possibility to communicate among the players, is introduced...
A bijection between 3-Motzkin paths and Schröder paths with no peak at odd.
A bijection between atomic partitions and unsplitable partitions.
A bijection between classes of fully packed loops and plane partitions.
A bijection between planar constellations and some colored Lagrangian trees.
A bijection on Dyck paths and its cycle structure.
A bijective proof of a major index theorem of Garsia and Gessel.
A bijective proof of Borchardt's identity.
A bijective proof of Garsia's -Lagrange inversion theorem.
A bijective proof of the hook-content formula for super Schur functions and a modified jeu de taquin.
A binomial coefficient identity associated to a conjecture of Beukers.