###### The Ryjáček Closure and a Forbidden Subgraph

## Abstract

The Ryjáček closure is a powerful tool in the study of Hamiltonian properties of claw-free graphs. Because of its usefulness, we may hope to use it in the classes of graphs defined by another forbidden subgraph. In this note, we give a negative answer to this hope, and show that the claw is the only forbidden subgraph that produces non-trivial results on Hamiltonicity by the use of the Ryjáček closure.

###### Bipartition Polynomials, the Ising Model, and Domination in Graphs

## Abstract

This paper introduces a trivariate graph polynomial that is a common generalization of the domination polynomial, the Ising polynomial, the matching polynomial, and the cut polynomial of a graph. This new graph polynomial, called the bipartition polynomial, permits a variety of interesting representations, for instance as a sum ranging over all spanning forests. As a consequence, the bipartition polynomial is a powerful tool for proving properties of other graph polynomials and graph invariants. We apply this approach to show that, analogously to the Tutte polynomial, the Ising polynomial introduced by Andrén and Markström in [3], can be represented as a sum over spanning forests.

###### 𝒫-Apex Graphs

## Abstract

Let 𝒫 be an arbitrary class of graphs that is closed under taking induced subgraphs and let 𝒞 (𝒫) be the family of forbidden subgraphs for 𝒫. We investigate the class 𝒫 (*k*) consisting of all the graphs *G* for which the removal of no more than *k* vertices results in graphs that belong to 𝒫. This approach provides an analogy to apex graphs and apex-outerplanar graphs studied previously. We give a sharp upper bound on the number of vertices of graphs in 𝒞 (𝒫 (1)) and we give a construction of graphs in 𝒞 (𝒫 (*k*)) of relatively large order for *k* ≥ 2. This construction implies a lower bound on the maximum order of graphs in 𝒞 (𝒫 (*k*)). Especially, we investigate 𝒞 (𝒲* _{r}*(1)), where 𝒲

*denotes the class of*

_{r}*P*-free graphs. We determine some forbidden subgraphs for the class 𝒲

_{r}*(1) with the minimum and maximum number of vertices. Moreover, we give sufficient conditions for graphs belonging to 𝒞(𝒫 (*

_{r}*k*)), where 𝒫 is an additive class, and a characterisation of all forests in C(𝒫 (

*k*)). Particularly we deal with 𝒞(𝒫 (1)), where 𝒫 is a class closed under substitution and obtain a characterisation of all graphs in the corresponding 𝒞(𝒫 (1)). In order to obtain desired results we exploit some hypergraph tools and this technique gives a new result in the hypergraph theory.

###### The Second Neighbourhood for Bipartite Tournaments

missing a generalized star , J. Graph Theory 71 (2012) 89–94. doi:10.1002/jgt.20634 [9] G. Gutin and R. Li, Seymour’s second neighbourhood conjecture for quasi-transitive oriented graphs . arxiv.org/abs/1704.01389. [10] F. Havet and S. Thomassé, Median orders of tournaments: a tool for the second neighborhood problem and Sumner’s conjecture , J. Graph Theory 35 (2000) 244–256. doi:10.1002/1097-0118(200012)35:4⟨244::AID-JGT2⟩3.0.CO;2-H [11] Y. Kaneko and S.C. Locke, The minimum degree approach for Paul Seymour’s distance 2 conjecture , Congr

###### Degree Sequences of Monocore Graphs

References [1] M. Altaf-Ul-Amin, K. Nishikata, T. Koma, T.Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, et al., Prediction of protein functions based on k-cores of protein-protein interaction networks and amino acid sequences, Genome Informatics 14 (2003) 498-499. [2] J. Alvarez-Hamelin, L. Dall’Asta, A. Barrat, A. Vespignani, k-core decomposition: a tool for the visualization of large scale networks, Adv. Neural Inf. Process. Syst. 18 (2006) 41. [3] G. Bader, C. Hogue, An automated method for finding

###### A Survey of the Path Partition Conjecture

. Anal. Issled. Oper. 9 (2002) 21-35 (in Russian). [46] L.S. Mel’nikov and I.V. Petrenko, Path kernels and cycle length in undirected graphs, in: V.N. Kassyanov (Ed.), Modern Problems of Program Construction, (ISI SB Russiam Academy of Science, Novosibirsk, 2002) 222-231 (in Russian). [47] L.S. Mel’nikov and I.V. Petrenko, Path kernels and partitions of graphs with small cycle length, In: Methods and Tools of Program Construction and Optimization, V.N. Kasyanov (Ed.), (ISI SB Russian Academy of Science, Novosibirsk, 2005) 145-160 (in

###### Relation Between Be-Algebras and G-Hilbert Algebras

## Abstract

Hilbert algebras are important tools for certain investigations in algebraic logic since they can be considered as fragments of any propositional logic containing a logical connective implication and the constant 1 which is considered as the logical value “true” and as a generalization of this was defined the notion of g-Hilbert algebra. In this paper, we investigate the relationship between g-Hilbert algebras, gi-algebras, implication gruopoid and BE-algebras. In fact, we show that every g-Hilbert algebra is a self distributive BE-algebras and conversely. We show cannot remove the condition self distributivity. Therefore we show that any self distributive commutative BE-algebras is a gi-algebra and any gi-algebra is strong and transitive if and only if it is a commutative BE-algebra. We prove that the MV -algebra is equivalent to the bounded commutative BE-algebra.

###### Idempotent Elements of Weak Projection Generalized Hypersubstitutions

## Abstract

A generalized hypersubstitution of type τ = (n_{i})_{i∈I} is a mapping σ which maps every operation symbol fi to the term σ (f_{i}) and may not preserve arity. It is the main tool to study strong hyperidentities that are used to classify varieties into collections called strong hypervarieties. Each generalized hypersubstitution can be extended to a mapping σ̂ on the set of all terms of type τ. A binary operation on Hyp_{G}(τ), the set of all generalized hypersubstitutions of type τ, can be defined by using this extension. The set Hyp_{G}(τ) together with such a binary operation forms a monoid, where a hypersubstitution σ_{id}, which maps f_{i} to f_{i}(x_{1}, . . . , x_{n₁} ) for every i ∈ I, is the neutral element of this monoid. A weak projection generalized hypersubstitution of type τ is a generalized hypersubstitution of type τ which maps at least one of the operation symbols to a variable. In semigroup theory, the various types of its elements are widely considered. In this paper, we present the characterizations of idempotent weak projection generalized hypersubstitutions of type (m, n) and give some sufficient conditions for a weak projection generalized hypersubstitution of type (m, n) to be regular, where m, n ≥ 1.

###### On the Gaussian Limiting Distribution of Lattice Points in a Parallelepiped

## Abstract

Let Γ ⊂ ℝ* ^{s}* be a lattice obtained from a module in a totally real algebraic number field. Let ℛ(

**,**

*θ***N**) be the error term in the lattice point problem for the parallelepiped [−

*θ*

_{1}

*N*

_{1},

*θ*

_{1}

*N*

_{1}] × . . . × [−

*θ*,

_{s}N_{s}*θ*]. In this paper, we prove that ℛ(

_{s}N_{s}**,**

*θ***N**)

*/σ*(ℛ,

**N**) has a Gaussian limiting distribution as

*N*→∞, where

**= (**

*θ**θ*

_{1}, . . . ,

*θ*) is a uniformly distributed random variable in [0, 1]

_{s}*,*

^{s}*N*=

*N*

_{1}. . . .

*N*and

_{s}*σ*(ℛ,

**N**) ≍ (log

*N*)

^{(s−1)/2}. We obtain also a similar result for the low discrepancy sequence corresponding to Γ. The main tool is the

*S*-unit theorem.

###### An Extremal Problem in Uniform Distribution Theory

Sequences. In: Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. [13] MCNEIL, A. J.—FREY, R.—EMBRECHTS, P.: Quantitative Risk Management. Princeton Series in Finance. (Concepts, techniques and tools) Princeton University Press, Princeton, NJ, 2005. [14] NELSEN, R. B.: An Introduction to Copulas, second edition. Springer Series in Statistics. Springer, New York, 2000. [15] PILLICHSHAMMER, F.—STEINERBERGER, S.: Average distance between consecutive points of uniformly distributed sequences, Unif