The research of the group focuses on interrelated aspects of combinatorics: Graph theory, Random Graphs, Probabilistic method, Geometric group theory and algebraic methods, Enumerative combinatorics, Combinatorial geometry, and Combinatorial number theory. Some recent research achievements are: Proof of the Maximum Distance Separable codes conjecture for prime fields; solution of a conjecture of Green on the removal lemma for systems of equations in finite fields

### Recent Submissions

• #### Grassl–Rötteler cyclic and consta-cyclic MDS codes are generalised Reed–Solomon codes ﻿

(Springer, 2022-12-31)
Article
Open Access
We prove that the cyclic and constacyclic codes constructed by Grassl and Rötteler in International Symposium on Information Theory (ISIT), pp. 1104–1108 (2015) are generalised Reed–Solomon codes. This note can be considered ...
• #### Irreducibility of the Tutte polynomial of an embedded graph ﻿

(2022-12-19)
Article
Open Access
We prove that the ribbon graph polynomial of a graph embedded in an orientable surface is irreducible if and only if the embedded graph is neither the disjoint union nor the join of embedded graphs. This result is analogous ...
• #### On additive MDS codes over small fields ﻿

(American Institute of Mathematical Sciences, 2022)
Article
Open Access
Let $C$ be a $(n,q^{2k},n-k+1)_{q^2}$ additive MDS code which is linear over ${\mathbb F}_q$. We prove that if $n \geq q+k$ and $k+1$ of the projections of $C$ are linear over ${\mathbb F}_{q^2}$ then $C$ ...
• #### Tutte uniqueness and Tutte equivalence ﻿

(Chapman & Hall/CRC, 2022-07-07)
Part of book or chapter of book
Restricted access - publisher's policy
This chapter considers graphs or matroids that have the same Tutte polynomial, as well as graphs or matroids that, up to isomorphism, are distinguished from all others by their Tutte polynomial. We call the former Tutte ...
• #### Positive Plücker tree certificates for non-realizability ﻿

(2022-01-24)
Article
Open Access
We introduce a new method for finding a non-realizability certificate of a simplicial sphere S: we exhibit a monomial combination of classical 3-term Pl¨ucker relations that yields a sum of products of determinants that ...
• #### Showing non-realizability of spheres by distilling a tree ﻿

(2021)
Conference report
Open Access
In [Zhe20a], Hailun Zheng constructs a combinatorial 3-sphere on 16 vertices whose graph is the complete 4-partite graph K4;4;4;4. Such a sphere seems unlikely to be realizable as the boundary complex of a 4-dimensional ...
• #### The multicolored graph realization problem ﻿

(Elsevier, 2022-07-01)
Article
Open Access
We introduce the multicolored graph realization problem (MGR). The input to this problem is a colored graph (G, φ), i.e., a graph G together with a coloring φ on its vertices. We associate each colored graph (G, φ) with a ...
• #### Polynomials for marked graphs and the chromatic symmetric function ﻿

Conference lecture
Restricted access - publisher's policy
• #### The equivalence of linear codes implies semi-linear equivalence ﻿

(2022-07-01)
Article
Open Access
We prove that if two linear codes are equivalent then they are semi-linearly equivalent. We also prove that if two additive MDS codes over a field are equivalent then they are additively equivalent.
• #### On the expected number of perfect matchings in cubic planar graphs ﻿

(2022-01-01)
Article
Open Access
A well-known conjecture by Lovász and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the ...