Fourier transforms and bent functions on faithful actions of finite abelian groups

Let G be a finite abelian group acting faithfully on a finite set X. The G-bentness and G-perfect nonlinearity of functions on X are studied by Poinsot and co-authors (Discret Appl Math 157:1848–1857, 2009; GESTS Int Trans Comput Sci Eng 12:1–14, 2005) via Fourier transforms of functions on G. In this paper we introduce the so-called \(G\)-dual set \(\widehat{X}\) of X, which plays the role similar to the dual group \(\widehat{G}\) of G, and develop a Fourier analysis on X, a generalization of the Fourier analysis on the group G. Then we characterize the bentness and perfect nonlinearity of functions on X by their own Fourier transforms on \(\widehat{X}\). Furthermore, we prove that the bentness of a function on X can be determined by its distance from the set of G-linear functions. As direct consequences, many known results in Logachev et al. (Discret Math Appl 7:547–564, 1997), Carlet and Ding (J Complex 20:205–244, 2004), Poinsot (2009), Poinsot et al. (2005) and some new results about bent functions on G are obtained. In order to explain the theory developed in this paper clearly, examples are also presented.     

Nonlinear functions and difference sets on group actions

There are many generalizations of the classical Boolean bent functions. Let G, H be finite groups and let X be a finite G-set. G-perfect nonlinear functions from X to H have been studied in several papers. They are generalizations of perfect nonlinear functions from G itself to H. By introducing the concept of a (G, H)-related difference family of X, we obtain a characterization of G-perfect nonlinear functions on X in terms of a (G, H)-related difference family. When G is abelian, we prove that there is a normalized G-dual set \(\widehat{X}\) of X, and characterize a G-difference set of X by the Fourier transform on a normalized G-dual set \({{\widehat{X}}}\). We will also investigate the existence and constructions of G-perfect nonlinear functions and G-bent functions. Several known results (IEEE Trans Inf Theory 47(7):2934–2943, 2001; Des Codes Cryptogr 46:83–96, 2008; GESTS Int Trans Comput Sci Eng 12:1–14, 2005; Linear Algebra Appl 452:89–105, 2014) are direct consequences of our results.     

Every 3-connected $$\{K_{1,3},N_{1,2,3}\}$$-free graph is Hamilton-connected

A graph G is \(\{X,Y\}\)-free if it contains neither X nor Y as an induced subgraph. Pairs of connected graphs X, Y such that every 3-connected \(\{X,Y\}\)-free graph is Hamilton-connected have been investigated recently in (2002, 2000, 2012). In this paper, it is shown that every 3-connected \(\{K_{1,3},N_{1,2,3}\}\)-free graph is Hamilton-connected, where \(N_{1,2,3}\) is the graph obtained by identifying end vertices of three disjoint paths of lengths 1, 2, 3 to the vertices of a triangle.     

Module operator virtual diagonals on the Fourier algebra of an inverse semigroup

For an amenable inverse semigroup S with the set of idempotents E and a minimal idempotent, we explicitly construct a contractive and positive module operator virtual diagonal on the Fourier algebra A(S), as a completely contractive Banach algebra and operator module over \(\ell ^1(E)\). This generalizes a well known result of Zhong-Jin Ruan on operator amenability of the Fourier algebra of a (discrete) group Ruan (Am J Math 117:1449–1474, 1995).     

Free groups and ends of graphs

Let X be a connected graph. An automorphism of X is said to be parabolic if it leaves no finite subset of vertices in X invariant and fixes precisely one end of X and hyperbolic if it leaves no finite subset of vertices in X invariant and fixes precisely two ends of X. Various questions concerning dynamics of parabolic and hyperbolic automorphisms are discussed.The set of ends which are fixed by some hyperbolic element of a group G acting on X is denoted by ?(G). If G contains a hyperbolic automorphism of X and G fixes no end of X, then G contains a free subgroup F such that ?(F) is dense in ?(G) with respect to the natural topology on the ends of X.As an application we obtain the following: A group which acts transitively on a connected graph and fixes no end has a free subgroup whose directions are dense in the end boundary.     

Snarks,Hypohamiltonian Graphs and Non-Supereulerian Graphs

A graph G is hypohamiltonian if it is not Hamiltonian but for each \(v\in V(G)\), the graph \(G-v\) is Hamiltonian. A graph is supereulerian if it has a spanning Eulerian subgraph. A graph G is called collapsible if for every even subset \(R\subseteq V(G)\), there is a spanning connected subgraph H of G such that R is the set of vertices of odd degree in H. A graph is reduced if it has no nontrivial collapsible subgraphs. In this note, we first prove that all hypohamiltonian cubic graphs are reduced non-supereulerian graphs. Then we introduce an operation to construct graphs from hypohamiltonian cubic graphs such that the resulting graphs are 3-edge-connected non-supereulerian reduced graphs and cannot be contracted to a snark. This disproves two conjectures, one of which was first posed by Catlin et al. in [Congr. Num. 76:173–181, 1990] and in [J. Combin. Theory, Ser B 66:123–139, 1996], and was posed again by Li et al. in [Acta Math. Sin. English Ser 30(2):291–304, 2014] and by Yang in [Supereulerian graphs, hamiltonicity of graphs and several extremal problems in graphs, Ph. D. Dissertation, Université Paris-Sub, September 27, 2013], respectively, the other one was posed by Yang 2013.     

Hamiltonian Type Properties in Claw-Free $$P_5$$-Free Graphs

A graph G on n vertices is said to be (k, m)-pancyclic if every set of k vertices in G is contained in a cycle of length r for each integer r in the set \(\{ m, m + 1, \ldots , n \}\). This property, which generalizes the notion of a vertex pancyclic graph, was defined by Faudree et al. in (Graphs Combin 20:291–310, 2004). The notion of (k, m)-pancyclicity provides one way to measure the prevalence of cycles in a graph. Broersma and Veldman showed in (Contemporary methods in graph theory, BI-Wiss.-Verlag, Mannheim, Wien, Zürich, pp 181–194, 1990) that any 2-connected claw-free \(P_5\)-free graph must be hamiltonian. In fact, every non-hamiltonian cycle in such a graph is either extendable or very dense. We show that any 2-connected claw-free \(P_5\)-free graph is (k, 3k)-pancyclic for each integer \(k \ge 2\). We also show that such a graph is (1, 5)-pancyclic. Examples are provided which show that these results are best possible. Each example we provide represents an infinite family of graphs.     

Hamilton Cycles in Implicit 2-Heavy Graphs

Let id(v) denote the implicit degree of a vertex v in a graph G. We define G of order n to be implicit 2-heavy if at least two of the end vertices of each induced claw have implicit degree at least \(\frac{n}{2}\). In this paper, we show that every implicit 2-heavy graph G is hamiltonian if we impose certain additional conditions on the connectivity of G or forbidden induced subgraphs. Our results extend two previous theorems of Broersma et al. (Discret Math 167–168:155–166, 1997) on the existence of Hamilton cycles in 2-heavy graphs.     

Pseudo-amenability of certain semigroup algebras

In this paper, we investigate the pseudo-amenability of semigroup algebra ? ¹(S), where S is an inverse semigroup with uniformly locally finite idempotent set. In particular, we show that for a Brandt semigroup \(S={\mathcal{M}}^{0}(G,I)\), the pseudo-amenability of ? ¹(S) is equivalent to the amenability of G.     

A Twelve Vertex Theorem for 3-Connected Claw-Free Graphs

The cyclability of a graph H, denoted by C(H), is the largest integer r such that H has a cycle through any r vertices. For a claw-free graph H, by Ryjá?ek (J Comb Theory Ser B 70:217–224, 1997) closure concept, there is a \(K_3\)-free graph G such that the closure \(cl(H)=L(G)\). In this note, we prove that for a 3-connected claw-free graph H with its closure \(cl(H)=L(G)\), \(C(H)\ge 12\) if and only if G can not be contracted to the Petersen graph in such a way that each vertex in P is obtained by contracting a nontrivial connected \(K_3\)-free subgraph. This is an improvement of the main result in Györi and Plummer (Stud Sci Math Hung 38:233–244, 2001).     

Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence

The Richardson variety X _α ^γ in the Grassmannian is defined to be the intersection of the Schubert variety X ^γ and opposite Schubert variety X _α . We give an explicit Gröbner basis for the ideal of the tangent cone at any T-fixed point of X _α ^γ , thus generalizing a result of Kodiyalam-Raghavan (J. Algebra 270(1):28–54, 2003) and Kreiman-Lakshmibai (Algebra, Arithmetic and Geometry with Applications, 2004). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the bounded RSK (BRSK). We use the Gröbner basis result to deduce a formula which computes the multiplicity of X _α ^γ at any T-fixed point by counting families of nonintersecting lattice paths, thus generalizing a result first proved by Krattenthaler (Sém. Lothar. Comb. 45:B45c, 2000/2001; J. Algebr. Comb. 22:273–288, 2005).     

Clique-Coloring Claw-Free Graphs

A clique is defined as a complete subgraph maximal under inclusion and having at least two vertices. A k-clique-coloring of a graph G is an assignment of k colors to the vertices of G such that no clique of G is monochromatic. Bacsó et al. (SIAM J Discrete Math 17:361–376, 2004) noted that the clique-coloring number is unbounded even for the line graphs of complete graphs. In this paper, we prove that a claw-free graph with maximum degree at most 7, except an odd cycle longer than 3, has a 2-clique-coloring by using a decomposition theorem of Chudnovsky and Seymour (J Combin Theory Ser B 98:839–938, 2008) and the limitation of the degree 7 is necessary since the line graph of \(K_{6}\) is a graph with maximum degree 8 but its clique-coloring number is 3 by the Ramsey number \(R(3,3)=6\). In addition, we point out that, if an arbitrary line graph of maximum degree at most d is m-clique-colorable (\(m\ge 3\)), then an arbitrary claw-free graph of maximum degree at most d is also m-clique-colorable.     

The Relaxed Edge-Coloring Game and <Emphasis Type="Italic">k</Emphasis>-Degenerate Graphs

The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with Δ(F)=Δ, then the first player, Alice, has a winning strategy for this game with r=Δ?j and d≥2j+2 for 0≤j≤Δ?1. This both improves and generalizes the result for trees in Dunn, C. (Discret. Math. 307, 1767–1775, 2007). More broadly, we generalize the main result in Dunn, C. (Discret. Math. 307, 1767–1775, 2007) by showing that if G is k-degenerate with Δ(G)=Δ and j∈[Δ+k?1], then there exists a function h(k,j) such that Alice has a winning strategy for this game with r=Δ+k?j and d≥h(k,j).     

A unifying approach to the Margolis–Meakin and Birget–Rhodes group expansion

Let G be a group. We show that the Birget–Rhodes prefix expansion \(G^{Pr}\) and the Margolis–Meakin expansion M(X; f) of G with respect to \(f:X\rightarrow G\) can be regarded as inverse subsemigroups of a common E-unitary inverse semigroup P. We construct P as an inverse subsemigroup of an E-unitary inverse monoid \(U/\zeta \) which is a homomorphic image of the free product U of the free semigroup \(X^+\) on X and G. The semigroup P satisfies a universal property with respect to homomorphisms into the permissible hull C(S) of a suitable E-unitary inverse semigroup S, with \(S/\sigma _S=G\), from which the characterizing universal properties of \(G^{Pr}\) and M(X; f) can be recaptured easily.     

Nordhaus–Gaddum Results for the Induced Path Number of a Graph When Neither the Graph Nor Its Complement Contains Isolates

The induced path number \(\rho (G)\) of a graph G is defined as the minimum number of subsets into which the vertex set of G can be partitioned so that each subset induces a path. A product Nordhaus–Gaddum-type result is a bound on the product of a parameter of a graph and its complement. Hattingh et al. (Util Math 94:275–285, 2014) showed that if G is a graph of order n, then \(\lceil \frac{n}{4} \rceil \le \rho (G) \rho (\overline{G}) \le n \lceil \frac{n}{2} \rceil \), where these bounds are best possible. It was also noted that the upper bound is achieved when either G or \(\overline{G}\) is a graph consisting of n isolated vertices. In this paper, we determine best possible upper and lower bounds for \(\rho (G) \rho (\overline{G})\) when either both G and \(\overline{G}\) are connected or neither G nor \(\overline{G}\) has isolated vertices.     

<Emphasis Type="Italic">C</Emphasis>*-Algebras Generated by Mappings. Criterion of Irreducibility

We study the operator algebra associated with a self-mapping ? on a countable set X which can be represented as a directed graph. The algebra is generated by the family of partial isometries acting on the corresponding l²(X). We study the structure of involutive semigroup multiplicatively generated by the family of partial isometries. We formulate the criterion when the algebra is irreducible on the Hilbert space. We consider the concrete examples of operator algebras. In particular, we give the examples of nonisomorphic C*-algebras, which are the extensions by compact operators of the algebra of continuous functions on the unit circle.     

Fixed point theorems for contractions of Reich type on a metric space with a graph

In a metric space with a directed graph G, Jachymski (Proc Am Math Soc 1(136):1359–1373, 2008) introduced the concept of Banach G-contraction and proved two fixed point theorems for such mappings. Bojor (Nonlinear Anal 75:3895–3901, 2012) generalized this concept to Reich G-contraction and obtain a fixed point theorem. Note that Bojor’s theorem is established under the additional type of connectedness of G and it does not include Jachymski’s results as a special case. Moreover, there are some mistakes in several corollaries. Some examples and counterexamples are illustrated. It is our purpose to improve Bojor’s theorem and to present two fixed point theorems for Reich G-contractions. Our results are extensions of the two Jachymski’s theorems. Finally, we also discuss some priori error estimates.     

Rigid Binary Relations on a 4-Element Domain

An n-ary relation ρ on a set U is strongly rigid if it is preserved only by trivial operations. It is projective if the only idempotent operations in P o l ρ are projections. Rosenberg, (Rocky Mt. J. Math. 3, 631–639, 1973) characterized all strongly rigid relations on a set with two elements and found a strongly rigid binary relation on every domain U of at least 3 elements. Larose and Tardif (Mult.-Valued Log. 7(5-6), 339–362, 2001) studied the projective and strongly rigid graphs and constructed large families of strongly rigid graphs. ?uczak and Ne?et?il (J. Graph Theory. 47, 81–86, 2004) settled in the affirmative a conjecture of Larose and Tardif that most graphs on a large set are projective, and characterized all homogenous graphs that are projective. ?uczak and Ne?et?il (SIAM J. Comput. 36(3), 835–843, 2006) confirmed a conjecture of Rosenberg that most relations on a big set are strongly rigid. In this paper, we characterize all strongly rigid relations on a set with at least three elements to answer an open question by Rosenberg, (Rocky Mt. J. Math. 3, 631–639, 1973) and we classify the binary relations on the 4-element domain by rigidity and demonstrate that there are merely 40 pairwise nonisomorphic rigid binary relations on the same domain (among them 25 are pairwise nonisomorphic strongly rigid).     

Critical Independent Sets and König–Egerváry Graphs

A set S of vertices is independent or stable in a graph G, and we write S ∈ Ind (G), if no two vertices from S are adjacent, and α(G) is the cardinality of an independent set of maximum size, while core(G) denotes the intersection of all maximum independent sets. G is called a König–Egerváry graph if its order equals α(G) + μ(G), where μ(G) denotes the size of a maximum matching. The number def (G) = | V(G) | ?2μ(G) is the deficiency of G. The number \({d(G)=\max\{\left\vert S\right\vert -\left\vert N(S)\right\vert :S\in\mathrm{Ind}(G)\}}\) is the critical difference of G. An independent set A is critical if \({\left\vert A\right\vert -\left\vert N(A)\right\vert =d(G)}\) , where N(S) is the neighborhood of S, and α _c (G) denotes the maximum size of a critical independent set. Larson (Eur J Comb 32:294–300, 2011) demonstrated that G is a König–Egerváry graph if and only if there exists a maximum independent set that is also critical, i.e., α _c (G) = α(G). In this paper we prove that: (i) \({d(G)=\left \vert \mathrm{core}(G) \right \vert -\left \vert N (\mathrm{core}(G))\right\vert =\alpha(G)-\mu(G)=def \left(G\right)}\) holds for every König–Egerváry graph G; (ii) G is König–Egerváry graph if and only if each maximum independent set of G is critical.     

Rainbow Connection in Some Digraphs

An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. This concept was introduced by Chartrand et al. (Math Bohemica 133(1):85–98, 2008), and it was extended to oriented graphs by Dorbec et al. (Discrete Appl Math 179(31):69–78, 2014). In this paper we present some results regarding this extension, mostly for the case of circulant digraphs.