A 1983 conjecture of Bouchet states that every flow-admissible signed graph has a nowhere-zero six-flow. We prove this conjecture for cyclically five-edge-connected, cubic signed graphs.
In this note, we prove an interesting result about perfect matchings in a complete bipartite graph with 2n vertices on each side, whose edges are colored in red and blue such that each vertex is part of n red edges and n blue edges.
A design is called $t$-pyramidal when it has an automorphism group which fixes $t$ points and acts sharply transitively on the remaining points. We determine all symmetric $(2^k-1,2^{k-1},2^{k-2})$-designs which are $(2^{k-1}-1)$-pyramidal over abelian groups.
We establish some bounds on the number of higher-dimensional partitions by volume. In particular, we give bounds via vector partitions and MacMahon's numbers.
Let F,G,H be three graphs on the same n vertices. We consider the maximum of the sum and product of the number of their edges subject to the condition in the title.
This article will prove a theorem for the existence of k-factor for k>1 ,and present an efficient algorithm for computing k-factor for all values of k based on this theorem.
The matching number of a graph G is the size of a maximum matching in the graph. In this note, we present a sufficient condition involving the matching number for the Hamiltonicity of graphs.
We give a common matroidal generalisation of `A Cantor-Bernstein theorem for paths in graphs' by Diestel and Thomassen and `A Cantor-Bernstein-type theorem for spanning trees in infinite graphs' by ourselves.
Consider the set Fn of factorizations of the full cycle (0 1 2 · · · n) ∈ S{0,1,...,n} into n transpositions. Write any such factorization (a1 b1) · · · (an bn) with all ai < bi to define its lower and upper sequences (a1, . . . , an) and (b1,...,bn), respectively. Remarkably, any factorization can be uniquely recovered from its lower (or upper) sequence. In fact, Biane (2002) showed that the simple map sending a factorization to its lower sequence is a bijection from Fn to the set Pn of parking functions of length n. Reversing this map to recover the factorization (and, hence, upper sequence) corresponding to a given lower sequence is nontrivial.
We give a different presentation of a recent bijection due to Chapuy and Dołe ̨ga for nonorientable bipartite quadrangulations and we extend it to the case of nonorientable general maps. This can be seen as a Bouttier–Di Francesco–Guitter-like generalization of the Cori–Vauquelin–Schaeffer bijection in the context of general nonori- entable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and we recover a famous asymptotic enumeration formula found by Gao.