Total Rainbow Connection Number and Complementary Graph

A vertex-colored graph G is rainbow vertex connected if any two distinct vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex connection number of G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex connected. In this paper, we prove that for a connected graph G, if \({{\rm diam}(\overline{G}) \geq 3}\), then \({{\rm rvc}(G) \leq 2}\), and this bound is tight. Next, we obtain that for a triangle-free graph \({\overline{G}}\) with \({{\rm diam}(\overline{G}) = 2}\), if G is connected, then \({{\rm rvc}(G) \leq 2}\), and this bound is tight. A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we prove that for a triangle-free graph \({\overline{G}}\) with \({{\rm diam}(\overline{G}) = 3}\), if G is connected, then trc\({(G) \leq 5}\), and this bound is tight. Next, a Nordhaus–Gaddum-type result for the total rainbow connection number is provided. We show that if G and \({\overline{G}}\) are both connected, then \({6 \leq {\rm trc} (G) + {\rm trc}(\overline{G}) \leq 4n - 6.}\) Examples are given to show that the lower bound is tight for \({n \geq 7}\) and n = 5. Tight lower bounds are also given for n = 4, 6.     

On $$\mathfrak {F}_{hq}$$-supplemented subgroups of a finite group

A subgroup H of a finite group G is quasinormal in G if it permutes with every subgroup of G. A subgroup H of a finite group G is \(\mathfrak {F}_{hq}\)-supplemented in G if G has a quasinormal subgroup N such that HN is a Hall subgroup of G and \((H\cap N)H_{G}/ H_{G} \le Z_{\mathfrak {F}}(G/H_{G})\), where \(H_{G}\) is the core of H in G and \({Z}_{\mathfrak {F}} (G/H_{G})\) is the \(\mathfrak {F}\)-hypercenter of \({G/H}_{G}\). This paper concerns the structure of a finite group G under the assumption that some subgroups of G are \(\mathfrak {F}_{hq}\)-supplemented in G.     

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.     

Geometry of Coadjoint Orbits and Multiplicity-one Branching Laws for Symmetric Pairs

Consider the restriction of an irreducible unitary representation π of a Lie group G to its subgroup H. Kirillov’s revolutionary idea on the orbit method suggests that the multiplicity of an irreducible H-module ν occurring in the restriction π|_H could be read from the coadjoint action of H on \(\mathcal {O}^{G} \cap \text {pr}^{-1}({\mathcal {O}}^{H})\), provided π and ν are ‘geometric quantizations’ of a G-coadjoint orbit \(\mathcal {O}^{G}\) and an H-coadjoint orbit \(\mathcal {O}^{H}\), respectively, where \(\text {pr} \colon \sqrt {-1}\mathfrak {g}^{\ast } \to \sqrt {-1}\mathfrak {h}^{\ast }\) is the projection dual to the inclusion \(\mathfrak {h} \subset \mathfrak {g}\) of Lie algebras. Such results were previously established by Kirillov, Corwin and Greenleaf for nilpotent Lie groups. In this article, we highlight specific elliptic orbits \(\mathcal {O}^{G}\) of a semisimple Lie group G corresponding to highest weight modules of scalar type. We prove that the Corwin–Greenleaf number \(\sharp (\mathcal {O}^{G} \cap \text {pr}^{-1}({\mathcal {O}}^{H}))/H\) is either zero or one for any H-coadjoint orbit \(\mathcal {O}^{H}\), whenever (G,H) is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits \(\mathcal {O}^{H}\) with nonzero Corwin–Greenleaf number. Our results coincide with the prediction of the orbit philosophy, and can be seen as ‘classical limits’ of the multiplicity-free branching laws of holomorphic discrete series representations (Kobayashi [Progr. Math. 2007]).     

Subgraph Transversal of Graphs

Given graphs F and G, denote by \({\tau_F}(G)\) the cardinality of a smallest subset \(T {\subseteq}V(G)\) that meets every maximal F-free subgraph of G. Erdös, Gallai and Tuza [9] considered the question of bounding \(\tau_{\overline{K_2}}(G)\) by a constant fraction of |G|. In this paper, we will give a complete answer to the following question: for which F, is τ _F (G) bounded by a constant fraction of |G|?In addition, for those graphs F for which \({\tau_F}(G)\) is not bounded by any fraction of |G|, we prove that \(\tau_F(G)\le|G|-\frac{1}{2}\sqrt{|G|}+\frac{1}{2}\), provided F is not K _k or \(\overline{K_k}\).     

Non-abelian tensor square of finite-by-nilpotent groups

Let G be a group. We denote by \({\nu(G)}\) an extension of the non-abelian tensor square \({G \otimes G}\) by \({G \times G}\). We prove that if G is finite-by-nilpotent, then the non-abelian tensor square \({G \otimes G}\) is finite-by-nilpotent. Moreover, \({\nu(G)}\) is nilpotent-by-finite (Theorem A). Also we characterize BFC-groups in terms of \({\nu(G)}\) among the groups G in which the derived subgroup is finitely generated (Theorem B).     

On Modular Edge-Graceful Graphs

Let G be a connected graph of order \({n\ge 3}\) and size m and \({f:E(G)\to \mathbb{Z}_n}\) an edge labeling of G. Define a vertex labeling \({f': V(G)\to \mathbb{Z}_n}\) by \({f'(v)= \sum_{u\in N(v)}f(uv)}\) where the sum is computed in \({\mathbb{Z}_n}\) . If f′ is one-to-one, then f is called a modular edge-graceful labeling and G is a modular edge-graceful graph. A graph G is modular edge-graceful if G contains a modular edge-graceful spanning tree. Several classes of modular edge-graceful trees are determined. For a tree T of order n where \({n\not\equiv 2 \pmod 4}\) , it is shown that if T contains at most two even vertices or the set of even vertices of T induces a path, then T is modular edge-graceful. It is also shown that every tree of order n where \({n\not\equiv 2\pmod 4}\) having diameter at most 5 is modular edge-graceful.     

A note on standard modules and Vogan L-packets

Let F be a non-Archimedean local field of characteristic 0, let G be the group of F-rational points of a connected reductive group defined over F and let \({G\prime}\) be the group of F-rational points of its quasi-split inner form. Given standard modules \({I(\tau, \nu )}\) and \({I(\tau\prime, \nu\prime)}\) for G and \({G\prime}\) respectively with \({\tau\prime}\) a generic tempered representation, such that the Harish-Chandra \({\mu}\)-function of a representation in the supercuspidal support of \({\tau}\) agrees with the one of a generic essentially square-integral representation in some Jacquet module of \({\tau\prime}\) (after a suitable identification of the underlying spaces under which \({\nu = \nu\prime}\)), we show that \({I(\tau, \nu)}\) is irreducible whenever \({I(\tau\prime, \nu\prime)}\) is. The conditions are satisfied if the Langlands quotients \({J(\tau, \nu})\) and \({J(\tau\prime, \nu\prime)}\) of respectively \({I(\tau, \nu)}\) and \({I(\tau\prime, \nu\prime)}\) lie in the same Vogan L-packet (whenever this Vogan L-packet is defined), proving that, for any Vogan L-packet, all the standard modules with Langlands quotient in a given Vogan L-packet are irreducible, if and only if this Vogan L-packet contains a generic representation. This result for generic Vogan L-packets was proven for quasi-split orthogonal and symplectic groups by Moeglin-Waldspurger and used in their proof of the general case of the local Gan-Gross-Prasad conjectures for these groups.     

Notes on the Kazhdan–Lusztig Theorem on Equivalence of the Drinfeld Category and the Category of $\boldsymbol{U}_{\!\boldsymbol q}{\boldsymbol {\mathfrak g}}$-Modules

We discuss the proof of Kazhdan and Lusztig of the equivalence of the Drinfeld category \({\mathcal D}({\mathfrak g},\hbar)\) of \({\mathfrak g}\)-modules and the category of finite dimensional \(U_q{\mathfrak g}\)-modules, \(q=e^{\pi i\hbar}\), for \(\hbar\in{\mathbb C}\setminus{\mathbb Q}^*\). Aiming at operator algebraists the result is formulated as the existence for each \(\hbar\in i{\mathbb R}\) of a normalized unitary 2-cochain \({\mathcal F}\) on the dual \(\hat G\) of a compact simple Lie group G such that the convolution algebra of G with the coproduct twisted by \({\mathcal F}\) is *-isomorphic to the convolution algebra of the q-deformation G _q of G, while the coboundary of \({\mathcal F}^{-1}\) coincides with Drinfeld’s KZ-associator defined via monodromy of the Knizhnik–Zamolodchikov equations.     

The limit case of a domination property

The domination number γ(G) of a connected graph G of order n is bounded below by(n+2-e(G))/ 3 , where (G) denotes the maximum number of leaves in any spanning tree of G. We show that (n+2-e(G))/ 3 = γ(G) if and only if there exists a tree T ∈ T ( G) ∩ R such that n1(T ) = e(G), where n1(T ) denotes the number of leaves of T1, R denotes the family of all trees in which the distance between any two distinct leaves is congruent to 2 modulo 3, and T (G) denotes the set composed by the spanning trees of G. As a consequence of the study, we show that if (n+2-e(G))/ 3 = γ(G), then there exists a minimum dominating set in G whose induced subgraph is an independent set. Finally, we characterize all unicyclic graphs G for which equality (n+2-e(G))/ 3= γ(G) holds and we show that the length of the unique cycle of any unicyclic graph G with (n+2-e(G))/ 3= γ(G) belongs to {4} ∪ {3 , 6, 9, . . . }.     

A characterization of the desarguesian and the Figueroa planes of order $$q^{3}$$

Let \(\Pi \) be a plane of order \(q^{3}\), \(q>2\), admitting \(G\cong PGL(3,q)\) as a collineation group. By Dempwolff (Geometriae Dedicata 18:101–112, 1985) the plane \(\Pi \) contains a G-invariant subplane \(\pi _{0}\) isomorphic to PG(2, q) on which G acts 2-transitively. In this paper it is shown that, if the homologies of \(\pi _{0}\) contained in G extend to \(\Pi \) then \(\Pi \) is either the desarguesian or the Figueroa plane.     

Disposition p-groups

For any prime p and positive integers c, d there is up to isomorphism a unique p-group \({G_{d}^{c}(p)}\) of least order having any (finite) p-group G with rank \({d(G) \le d}\) and Frattini class \({c_{p}(G) \le c}\) as epimorphic image. Here \({c_{p}(G) = n}\) is the least positive integer such that G has a central series of length n with all factors being elementary. This “disposition” p-group \({G_{d}^{c}(p)}\) has been examined quite intensively in the literature, sometimes controversially. The objective of this paper is to present a summary of the known facts, and to add some new results. For instance we show that for \({G = G_{d}^{c}(p)}\) the centralizer \({C_{G}(x) = \langle Z(G), x \rangle}\) whenever \({x \in G}\) is outside the Frattini subgroup, and that for odd p and \({d \ge 2}\) the group \({E = G_{d}^{c+1}(p)/(G_{d}^{c+1}(p))^{p^{c}}}\) is a distinguished Schur cover of G with \({E/Z(E) \cong G}\). We also have a fibre product construction of \({G_{d}^{c+1}(p)}\) in terms of \({G = G_{d}^{c}(p)}\) which might be of interest for Galois theory.     

On a class of finite capable <Emphasis Type="Italic">p</Emphasis>-groups

A group G is called capable if there is a group H such that \({G \cong H/Z(H)}\) is isomorphic to the group of inner automorphisms of H. We consider the situation that G is a finite capable p-group for some prime p. Suppose G has rank \({d(G) \ge 2}\) and Frattini class \({c \ge 1}\), which by definition is the length of a shortest central series of G with all factors being elementary abelian. There is up to isomorphism a unique largest p-group \({G_d^c}\) with rank d and Frattini class c, and G is an epimorphic image of \({G_d^c}\). We prove that this \({G_d^c}\) is capable; more precisely, we have \({G_d^c \cong G_d^{c+1}/Z(G_d^{c+1})}\).     

Depth,Stanley depth,and regularity of ideals associated to graphs

Let \({\mathbb{K}}\) be a field and \({S=\mathbb{K}[x_1,\dots,x_n]}\) be the polynomial ring in n variables over \({\mathbb{K}}\). Let G be a graph with n vertices. Assume that \({I=I(G)}\) is the edge ideal of G and \({J=J(G)}\) is its cover ideal. We prove that \({{\rm sdepth}(J)\geq n-\nu_{o}(G)}\) and \({{\rm sdepth}(S/J)\geq n-\nu_{o}(G)-1}\), where \({\nu_{o}(G)}\) is the ordered matching number of G. We also prove the inequalities \({{\rmsdepth}(J^k)\geq {\rm depth}(J^k)}\) and \({{\rm sdepth}(S/J^k)\geq {\rmdepth}(S/J^k)}\), for every integer \({k\gg 0}\), when G is a bipartite graph. Moreover, we provide an elementary proof for the known inequality reg\({(S/I)\leq \nu_{o}(G)}\).     

Weak regularity of group algebras

A Banach algebra \(\mathcal {A}\) is called weakly regular if its multiplicative semigroup is E-inversive. We show that for a unimodular group G which admits an integrable unitary representation, \(L^1(G)\) is weakly regular. Moreover for a locally compact Abelian group, \(L^1(G)\) is weakly regular if and only if G is compact; while \(L^1(G)^{**}\) is weakly regular if and only if G is finite. All of our results hold, if we replace \(L^1(G)\) with M(G).     

Group theoretic characterizations of Buekenhout–Metz unitals in $\mathop{\mathrm{PG}}(2,q^{2})$

Let G be the group of projectivities stabilizing a unital \(\mathcal{U}\) in \(\mathop{\mathrm{PG}}(2,q^{2})\) and let A,B be two distinct points of \(\mathcal{U}\). In this paper we prove that, if G has an elation group of order q with center A and a group of projectivities stabilizing both A and B of order a divisor of q?1 greater than \(2(\sqrt{q}-1)\), then \(\mathcal{U}\) is an ovoidal Buekenhout–Metz unital. From this result two group theoretic characterizations of orthogonal Buekenhout–Metz unitals are given.     

On color-preserving automorphisms of Cayley graphs of odd square-free order

An automorphism \(\alpha \) of a Cayley graph \(\mathrm{Cay}(G,S)\) of a group G with connection set S is color-preserving if \(\alpha (g,gs) = (h,hs)\) or \((h,hs^{-1})\) for every edge \((g,gs)\in E(\mathrm{Cay}(G,S))\). If every color-preserving automorphism of \(\mathrm{Cay}(G,S)\) is also affine, then \(\mathrm{Cay}(G,S)\) is a Cayley color automorphism (CCA) graph. If every Cayley graph \(\mathrm{Cay}(G,S)\) is a CCA graph, then G is a CCA group. Hujdurovi? et al. have shown that every non-CCA group G contains a section isomorphic to the non-abelian group \(F_{21}\) of order 21. We first show that there is a unique non-CCA Cayley graph \(\Gamma \) of \(F_{21}\). We then show that if \(\mathrm{Cay}(G,S)\) is a non-CCA graph of a group G of odd square-free order, then \(G = H\times F_{21}\) for some CCA group H, and \(\mathrm{Cay}(G,S) = \mathrm{Cay}(H,T)\mathbin {\square }\Gamma \).     

Families of conditionally approximately convex functions

Let \((G,+)\) be an Abelian topological group, which is also a \(T_{0}\)-space and a Baire space simultaneously, D be an open connected subset of G and \(\alpha : D-D \rightarrow {\mathbb R}\) be a function continuous at zero and such that \(\alpha (0)=0\). We show that if \((f_n)\) is a sequence of continuous functions \(f_n : D \rightarrow {\mathbb R}\) such that \(f_n(z) \le \frac{1}{2} f_n(x)+\frac{1}{2}f(y)+\alpha (x-y)\) for \(n\in {\mathbb N}\) and \(x,y,z\in D\) such that \(2z=x+y\) and if \((f_n)\) is pointwise convergent [bounded] then it is convergent uniformly on compact subsets of D [in the case when G is additionally a separable space, it contains a subsequence which is convergent on compact subsets of D].     

A note on a result of Guo and Isaacs about p-supersolubility of finite groups

In this note, global information about a finite group is obtained by assuming that certain subgroups of some given order are S-semipermutable. Recall that a subgroup H of a finite group G is said to be S-semipermutable if H permutes with all Sylow subgroups of G of order coprime to \({\lvert H\rvert}\). We prove that for a fixed prime p, a given Sylow p-subgroup P of a finite group G, and a power d of p dividing \({\lvert G\rvert}\) such that \({1\le d < \lvert P\rvert}\), if \({H\,{\cap}\, O^p(G)}\) is S-semipermutable in \({O^p(G)}\) for all normal subgroups H of P with \({\lvert H\rvert=d}\), then either G is p-supersoluble or else \({\lvert P\,{\cap}\, {O^p(G)}\rvert > d}\). This extends the main result of Guo and Isaacs in (Arch. Math. 105:215–222 2015). We derive some theorems that extend some known results concerning S-semipermutable subgroups.