Aequationes mathematicae

A family of 4-dimensional pseudomanifolds is introduced using a standard graph-theoretical representation of lens spaces Some homeomorphisms between these “lens-like” spaces are established, the computation of their fundamental groups and of bounds for their genera are carried out

The Gárfás–Sumner conjecture asks whether for every tree T, the class of (induced) T-free graphs is $$\chi $$ -bounded. The conjecture is solved for several special trees, but it is still open in general. Motivated by the conjecture, the chromatic number of triangle-free and broom-free graphs is well studied, since a broom is one of the generalizations of a star, where a broom B(m, n) is the graph obtained from a star $$K_{1,n}$$ and an m-vertex path $$P_m$$ by identifying the center of $$K_{1,n}$$ and a leaf of $$P_m$$ . Gárfás, Szemeredi and Tuza proved that for every triangle-free and B(m, n)-free graph G, $$\chi (G) \le m+n-1$$ . This upper bound has been improved by Wang and Wu to $$m+n-2$$ for $$m\ge 2, n\ge 1$$ . In this paper, we prove that any triangle-free and B(4, 2)-free graph G is 3-colorable if the number of vertices of G is at least 17. Furthermore, the above estimation is the best possible since there exists a triangle-free and B(4, 2)-free 4-chromatic graph with sixteen vertices, named the Clebsch graph.

The general solutions of the functional equations $$\begin{aligned} F(pq)= & {} q^\alpha G(p)+p^\alpha H(q)+K(p)L(q),\\ F(pq)= & {} q^\beta G(p)+p^\alpha H(q)+cK(p)K(q) \end{aligned}$$ and $$\begin{aligned} f(pq)=q^\alpha f(p)+p^\alpha f(q) +cg(p)g(q) \end{aligned}$$ with $$g(1)=0$$ and c, a given nonzero real constant, are obtained. Here F, G, H, K, L, f and g are real-valued functions each with domain I, the unit closed interval and $$1\ne \alpha >0$$ , $$\alpha \in {\mathbb {R}}$$ ; $$1\ne \beta >0$$ , $$\beta \in {\mathbb {R}}$$ .

The functional equation related to competition ([2]) $$f\left( \frac{x+y}{1-xy}\right) =\frac{f\left( x\right) +f\left(y\right)} {1+f\left( x\right) f\left( y\right)},\qquad x,y\in\mathbb{R}, xy\neq 1,$$ for y = cx with a fixed c > 0, leads to the equation $$f\left( \frac{\left( 1+c\right) x}{1-cx^{2}}\right) =\frac{f\left(x\right) +f\left( cx\right)} {1+f\left( x\right) f\left( cx\right)},\qquad x\in \mathbb{R}, \left\vert x \right\vert <\frac{1}{\sqrt{c}}.$$ The case c = 1 (a first order iterative functional equation) was treated in [3]. In this paper we consider the case c ≠ 1 (when the equation is of the second order). We show that a function $${f:\mathbb{R} \rightarrow \mathbb{R},\,f\left( 0\right) =0}$$ , differentiable at the point 0 satisfies this functional equation iff there is a real p such that $${f=\tanh \circ \left( p\tan ^{-1} \right) }$$ which extends the main result of [3].

Cauchy means are defined as those obtained from applying the Cauchy mean value theorem to a pair of suitable functions. A so-called mixing operator is defined, in such a way that each Cauchy mean is the unique fixed point of the mixing operator associated with two quasiarithmetic means. As a consequence, some comparison results for Cauchy means are obtained, together with a functional characterization relating a subclass of Cauchy means, the so-called anti-Lagrangian means, with quasiarithmetic ones.

Convergence in category of a sequence of real-valued functions has been introduced by E. Wagner as an analogue of convergence in measure. In the paper it is shown that in some circumstances both kinds of convergence behave similarly, but sometimes the behaviour is different.

In the original publication, Example 2.2 was incorrectly published.