39 research outputs found
Neighborhood intersections and Hamiltonicity in almost claw-free graphs
AbstractLet G be a graph. The partially square graph G∗ of G is a graph obtained from G by adding edges uv satisfying the conditions uv∉E(G), and there is some w∈N(u)∩N(v), such that N(w)⊆N(u)∪N(v)∪{u,v}. Let t>1 be an integer and Y⊆V(G), denote n(Y)=|{v∈V(G)|miny∈Y{distG(v,y)}⩽2}|,It(G)={Z|Z is an independent set of G,|Z|=t}. In this paper, we show that a k-connected almost claw-free graph with k⩾2 is hamiltonian if ∑z∈Zd(z)⩾n(Z)−k in G for each Z∈Ik+1(G∗), thereby solving a conjecture proposed by Broersma, Ryjác̆ek and Schiermeyer. Zhang's result is also generalized by the new result
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Discrete Geometry (hybrid meeting)
A number of important recent developments in various branches of
discrete geometry were presented at the workshop, which took place in
hybrid format due to a pandemic situation. The presentations
illustrated both the diversity of the area and its strong connections
to other fields of mathematics such as topology, combinatorics,
algebraic geometry or functional analysis. The open questions abound
and many of the results presented were obtained by young researchers,
confirming the great vitality of discrete geometry
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Combinatorics
Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing abstracts of the presentations and a summary of the problem session