Sierpiński graphs as spanning subgraphs of Hanoi graphs

Hanoi graphs H _p ⁿ model the Tower of Hanoi game with p pegs and n discs. Sierpinski graphs S _p ⁿ arose in investigations of universal topological spaces and have meanwhile been studied extensively. It is proved that S _p ⁿ embeds as a spanning subgraph into H _p ⁿ if and only if p is odd or, trivially, if n = 1.     

Group connectivity and group colorings of graphs — A survey

In 1950s, Tutte introduced the theory of nowhere-zero flows as a tool to investigate the coloring problem of maps, together with his most fascinating conjectures on nowhere-zero flows. These have been extended by Jaeger et al. in 1992 to group connectivity, the nonhomogeneous form of nowhere-zero flows. Let G be a 2-edge-connected undirected graph, A be an (additive) abelian group and A* = A − {0}. The graph G is A-connected if G has an orientation D(G) such that for every map b: V (G) ↦ A satisfying Σ _v∈V(G) b(v) = 0, there is a function f: E(G) ↦ A* such that for each vertex v ∈ V (G), the total amount of f-values on the edges directed out from v minus the total amount of f-values on the edges directed into v is equal to b(v). The group coloring of a graph arises from the dual concept of group connectivity. There have been lots of investigations on these subjects. This survey provides a summary of researches on group connectivity and group colorings of graphs. It contains the following sections.

Small topological complete subgraphs of “dense” graphs

A graph ofn vertices and edges contains aTK _t on at most 7t ² logt/ vertices. This answers a question of P. Erds.     

Stability in vector maximization—A survey

Stability and sensitivity analysis becomes more and more attractive also in the area of multicriteria decision making (MCDM). Influences of parameter changes concerning the right-hand-side or the objective functions respectively or changes of the domination structure on the solution of various mathematical models of vectormaximization problems — as developed in several publications — are examined in this paper and a survey on the corresponding results is given. The survey is subdivided with respect to the various model structures of the decision problem — i.e. linear or nonlinear, deterministic or stochastic, static or dynamic — and with respect to the above mentioned investigated changes.     

A survey of the Szegő equation

Science China Mathematics - We survey the main properties of the cubic Szegő equation from the PDE viewpoint, emphasising global existence of smooth solutions, analytic regularity, growth of...     

Depleting the signal: Analysis of chemotaxis-consumption models—A survey

We give an overview of analytical results concerned with chemotaxis systems where the signal is absorbed. We recall results on existence and properties of solutions for the prototypical chemotaxis-consumption model and various variants and review more recent findings on its ability to support the emergence of spatial structures.     

Models of stochastic geometry — A survey

This paper discusses some models of stochastic geometry which are of potential interest for operations research. These are the Boolean model, a certain model for random compact sets and marked point processes. The Boolean model is a generalization of the well-known queueing systemM/G/. The random compact set model may be useful for modelling spatial spreading processes such as fires, cancers or holes in the Earth's surface. Marked point processes are used here as models of forests and used for a statistical study of the spatial distribution of damaged trees.Extended version of an Invited Lecture on the 16th Symposium for OR in Hamburg 1992.     

Topological minors in line graphs — A proof of Zha’s conjecture

In 1992, Xiaoya Zha conjectured that the line graph of a 3-connected non-planar graph contains a subdivision of K ₅. In this paper we prove this conjecture. This result is the main ingredient of [4] where a complete characterization of all the 4-connected claw-free graphs not containing a subdivision of K ₅ is obtained.     

A reconsideration of some embedded Runge—Kutta formulae

The RK5(4) and RK6(5) embedded Runge—Kutta formulae are reconsidered with regard to enlarging regions of absolute stability while retaining satisfactory truncation error norms. Results from standard tests for the above pairs are presented in comparison with an efficient RK8(7) embedded formula.     

Quantum cohomology of the Hilbert scheme of points in the plane

We determine the ring structure of the equivariant quantum cohomology of the Hilbert scheme of points of ℂ². The operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum Calogero-Sutherland many-body system. A relationship between the quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for local curves is proven.     

Uniformization of Sierpiński carpets in the plane

Let S _i , i∈I, be a countable collection of Jordan curves in the extended complex plane \(\widehat{\mathbb{C}}\) that bound pairwise disjoint closed Jordan regions. If the Jordan curves are uniform quasicircles and are uniformly relatively separated, then there exists a quasiconformal map \(f\colon\widehat{\mathbb{C}}\rightarrow\widehat{\mathbb{C}}\) such that f(S _i ) is a round circle for all i∈I. This implies that every Sierpiński carpet in \(\widehat{\mathbb{C}}\) whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round Sierpiński carpet by a quasisymmetric map.     

A graph-theoretical representation of PL-manifolds — A survey on crystallizations

This is a survey of the techniques and results developed by M. Pezzana and his group, which includes, besides the authors, A. Cavicchioli, P. Bandieri and A Donati.The original concept is that of contracted triangulation, which was introduced with the main goal of finding a minimal atlas for topological manifolds ([P₁ 1968], [P₂ 1974], [P₃ 1974], [FG₂ 1979]). Only later did the possibility of deducing a graph-theoretical tool — the crystallization — for representing P.L. manifolds occur as a major aspect of the theory ([P₄ 1975], [F₁ 1976]). This leads to an application of graph theory to P.L. topology, which seems not to have been explored before. Recently, other authors outside Italy have independently become interested in this subject.For the sake of conciseness, definitions and statements often appear in a form other than that of the quoted references.     

Results of the maximum genus of graphs

In this paper,we provide a new class of up-embeddable graphs,and obtain a tight lower bound on the maximum genus of a class of 2-connected pseudographs of diameter 2 and of a class of diameter 4 multi-graphs.This extends a result of Skoviera.     

Optimization problems of the third edge-connectivity of graphs

The third edge-connectivity λ3(G) of a graph G is defined as the minimum cardinality over all sets of edges, if any, whose deletion disconnects G and each component of the resulting graph has at least 3 vertices. An upper bound has been established for λ3(G) whenever λ3(G) is well-defined. This paper first introduces two combinatorial optimization concepts, that is, maximality and superiority, of λ3(G), and then proves the Ore type sufficient conditions for G to be maximally and super third edge-connected. These concepts and results are useful in network reliability analysis.     

A survey of χ-boundedness

If a graph has bounded clique number and sufficiently large chromatic number, what can we say about its induced subgraphs? András Gyárfás made a number of challenging conjectures about this in the early 1980s, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now.     

On the adjacent-vertex-strongly-distinguishing total coloring of graphs

For any vertex u∈V(G), let T_N(U)={u}∪{uv|uv∈E(G), v∈v(G)}∪{v∈v(G)|uv∈E(G)}and let f be a total k-coloring of G. The total-color neighbor of a vertex u of G is the color set C_f(u)={f(x)|x∈TN(U)}. For any two adjacent vertices x and y of V(G)such that C_f(x)≠C_f(y), we refer to f as a k-avsdt-coloring of G("avsdt"is the abbreviation of"adjacent-vertex-strongly- distinguishing total"). The avsdt-coloring number of G, denoted by X_(ast)(G), is the minimal number of colors required for a avsdt-coloring of G. In this paper, the avsdt-coloring numbers on some familiar graphs are studied, such as paths, cycles, complete graphs, complete bipartite graphs and so on. We proveΔ(G) 1≤X_(ast)(G)≤Δ(G) 2 for any tree or unique cycle graph G.     

On Bernstein–Walsh-type lemmas in regions of the complex plane

Let G ì \mathbb C G \subset {\mathbb C} be a finite region bounded by a Jordan curve L: = ?G L: = \partial G , let W: = \textext[`(G)] \Omega : = {\text{ext}}\bar{G} (with respect to [`(\mathbb C)] {\overline {\mathbb C}} ), $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} , and let w = F(z) w = \Phi (z) be a univalent conformal mapping of Ω onto Δ normalized by $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 . By A _p (G); p > 0; we denote a class of functions f analytic in G and satisfying the condition