The chromatic vertex (resp. edge) stability number vsχ(G) esχ(G)) of a graph G is the minimum vertices edges) whose deletion results in H with χ(H)=χ(G)−1. In main result it proved that if χ(G)∈{Δ(G),Δ(G)+1}, then vsχ(G)=ivsχ(G), where ivsχ(G) independent number. need not hold for graphs χ(G)≤Δ(G)+12. It χ(G)>Δ(G)2+1, vsχ(G)=esχ(G). A Nordhaus–Gaddum-type on also given.

Abstract The aim of the current study was to analyze players’ interactions in serve and return situations examine way points were decided due positions surfaces. Fourteen ATP 1000 tournament matches analyzed (7 on hard surface; 7 clay surface). type surface did not affect ended (χ²=9.26; p > 0.05) but influence vertical positioning (χ²=270.86; < 0.001). positioning, both lateral (χ²=92.68...

A class G of graphs is said to be χ-bounded if there is a function f : N → R such that for all G ∈ G and all induced subgraphs H of G, χ(H) ≤ f(ω(H)). In this paper, we show that if G is a χ-bounded class, then so is the closure of G under any one of the following three operations: substitution, gluing along a clique, and gluing along a bounded number of vertices. Furthermore, if G is χ-bounded...

Given data from a spherical Gaussian distribution with unknown mean vector θ, estimates of quadratic functionals are constructed by thresholding. Mean squared error bounds are derived via a comparison with those already available for a suitable noncentral χ variate. By way of illustration, the resulting inequalities are used to yield an optimal rate adaptivity result for estimation of integrate...