Mathematics

Suppose a committee consisting of three members has to match n candidates to n different positions. Each member of the committee proposes a matching, however the proposed matchings totally disagree, i.e., every candidate is matched to three different positions according to three committee members. All three committee members are very competitive and want to push through as many of their suggestions as possible. Can a committee always find a compromise -- a matching of candidates to positions such that for every committee member a third of all candidates are assigned according to that committee member suggestion? We will consider an asymptotic version of this question and several other variants of similar problem. As an application we will consider an embedding problem -- in particular which configurations large Steiner systems always need to contain.

For k?? , a k -colouring c of G is a mapping from V(G) to {1,2,??k} such that c(u)?�c(v) for any two non-adjacent vertices u and v . The k -Colouring problem is to decide if a graph G has a k -colouring. For a family of graphs H , a graph G is H -free if G does not contain any graph from H as an induced subgraph. Let C s be the s -vertex cycle. In previous work (MFCS 2019) we examined the effect of bounding the diameter on the complexity of 3 -Colouring for ( C 3 ,?? C s ) -free graphs and H -free graphs where H is some polyad. Here, we prove for certain small values of s that 3 -Colouring is polynomial-time solvable for C s -free graphs of diameter 2 and ( C 4 , C s ) -free graphs of diameter 2 . In fact, our results hold for the more general problem List 3 -Colouring. We complement these results with some hardness result for diameter 4 .

In 1985 Crapo introduced in \cite{Crapo} a new mathematical object that he called \textit{geometry of circuits}. Four years later, in 1989, Manin and Schechtman defined in \cite{MS} the same object and called it \textit{discriminantal arrangement}, the name by which it is known now a days. Subsequently in 1997 Bayer and Brandt ( see \cite{BB} ) distinguished two different type of those arrangements calling \textit{very generic} the ones having intersection lattice of maximum cardinality and \textit{non very generic} the others. Results on the combinatorics of very generic arrangements already appear in Crapo \cite{Crapo} and in 1997 in Athanasiadis \cite{Atha} while the first known result on non very generic case is due to Libgober and the first author in 2018. In their paper \cite{LS}, they studied the combinatorics of non very generic case in rank 2 . In this paper we further develop their result providing a sufficient condition for the discriminantal arrangement to be non very generic which holds in rank r?? .

Let D be a k -regular bipartite tournament on n vertices. We show that, for every p with 2?�p?�n/2?? , D has a cycle C of length 2p such that D?�C is hamiltonian unless D is isomorphic to the special digraph F 4k . This statement was conjectured by Manoussakis, Song and Zhang [K. Zhang, Y. Manoussakis, and Z. Song. Complementary cycles containing a fixed arc in diregular bipartite tournaments. Discrete Mathematics, 133(1-3):325--328,1994]. In the same paper, the conjecture was proved for p=2 and more recently Bai, Li and He gave a proof for p=3 [Y. Bai, H. Li, and W. He. Complementary cycles in regular bipartite tournaments. Discrete Mathematics, 333:14--27, 2014].

We prove that the complement of a non-separating planar graph of order at least nine is intrinsically linked. We also prove that the complement of a non-separating planar graph of order at least 10 is intrinsically knotted. We show these lower bounds on the orders are the best possible. We show that for a maximal non-separating planar graph with n?? vertices, its complement cG is (n??)??apex. We conclude that the de Verdière invariant for such graphs satisfies μ(cG)?�n?? .

The nth r-extended Lah-Bell number is defined as the number of ways a set with n+r elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete r-extended Lah-Bell polynomials and complete r -extended Lah-Bell polynomials respectively as multivariate versions of r -Lah numbers and the r-extended Lah-Bell numbers and to investigate some properties and identities for these polynomials. From these investigations, we obtain some expressions for the r-Lah numbers and the r-extended Lah-Bell numbers as finite sums.

The dichromatic number ? ??(D) of a digraph D is the smallest k for which it admits a k -coloring where every color class induces an acyclic subgraph. Inspired by Hadwiger's conjecture for undirected graphs, several groups of authors have recently studied the containment of directed graph minors in digraphs with given dichromatic number. In this short note we improve several of the existing bounds and prove almost linear bounds by reducing the problem to a recent result of Postle on Hadwiger's conjecture.

Given a connected graph G=(V(G),E(G)) , the length of a shortest path from a vertex u to a vertex v is denoted by d(u,v) . For a proper subset W of V(G) , let m(W) be the maximum value of d(u,v) as u ranging over W and v ranging over V(G)?�W . The proper subset W={ w 1 ,?? w |W| } is a {\em completeness-resolving set} of G if Ψ W :V(G)?�W?�[m(W) ] |W| ,u??d( w 1 ,u),??d( w |W| ,u)) is a bijection, where [m(W) ] |W| ={( a (1) ,?? a (|W|) )????a (i) ?�m(W) for each i=1,??|W|}. A graph is {\em completeness-resolvable} if it admits a completeness-resolving set. In this paper, we first construct the set of all completeness-resolvable graphs by using the edge coverings of some vertices in given bipartite graphs, and then establish posets on some subsets of this set by the spanning subgraph relationship. Based on each poset, we find the maximum graph and give the lower and upper bounds for the number of edges in a minimal graph. Furthermore, minimal graphs satisfying the lower or upper bound are characterized.

In recent work, G. E. Andrews and G. Simay prove a surprising relation involving parity palindromic compositions, and ask whether a combinatorial proof can be found. We extend their results to a more general class of compositions that are palindromic modulo m , that includes the parity palindromic case when m=2 . We then provide combinatorial proofs for the cases m=2 and m=3 .

We construct for the first time conditionally decomposable d -polytopes for d?? . These examples have any number of vertices from 4d?? upwards. A polytope is said to be conditionally decomposable if one polytope combinatorially equivalent to it is decomposable (with respect to the Minkowski sum) and another one combinatorially equivalent to it is indecomposable. In our examples, one has 4d?? vertices and is the sum of a line segment and a bi-pyramid over a prism. The other ones have 4d?? vertices, and one of them is the sum of a line segment and a 2 -fold pyramid over a prism. We show that the latter examples have the minimum number of vertices among all conditionally decomposable d -polytopes that have a line segment for a summand.