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Posplošitev medianskih grafov: k-medianski grafi
No full textMedian graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. To be more formal, a graph G is a median graph if, for all μ, u, v ∈ V(G), it holds that |I(μ, u) ∩ I(μ, v) ∩ I(u, v)| = 1 where I(x, y) denotes the set of all vertices that lie on shortest paths connecting x and y.
In this paper we are interested in a natural generalization of median graphs, called k-median graphs. A graph G is a k-median graph, if there are k vertices μ1, …, μk ∈ V(G) such that, for all u, v ∈ V(G), it holds that |I(μ_i, u) ∩ I(μ_i, v) ∩ I(u, v)| = 1, 1 ≤ i ≤ k. By definition, every median graph with n vertices is an n-median graph. We provide several characterizations of k-median graphs that, in turn, are used to provide many novel characterizations of median graphs
k-dominacijske invariante Kneserjevih grafov
No full textIn this follow-up to work of M.G. Cornet and P. Torres from 2023, where the k-tuple domination number and the 2-packing number in Kneser graphs K(n, r) were studied, we are concerned with two variations, the k-domination number, γ_k(K(n, r)), and the k-tuple total domination number,
γ_{t × k}(K(n, r)), of K(n, r). For both invariants we prove monotonicity results by showing that γ_k(K(n, r)) ≥ γ_k(K(n + 1, r)) holds for any n ≥ 2(k + r), and γ_{t × k}(K(n, r)) ≥
γ_{t × k}(K(n + 1, r)) holds for any n ≥ 2r + 1. We prove that γ_k(K(n, r)) = γ_{t × k}(K(n, r)) = k + r when n ≥ r(k + r), and that in this case every γ_(k)-set and γ_(t × k)-set is a clique, while γ_k(r(k + r) − 1, r) = γ_{t × k}(r(k + r) − 1, r) = k + r + 1, for any k ≥ 2. Concerning the 2-packing number, ρ₂(K(n, r)), of K(n, r), we prove the exact values of ρ₂(K(3r − 3, r)) when r ≥ 10, and give sufficient conditions for ρ₂(K(n, r)) to be equal to some small values by imposing bounds on r with respect to n. We also prove a version of monotonicity for the 2-packing number of Kneser graphs
Bollobáseve neenakosti para množic za sestave
No full textA d-composition of a set S is an ordered d-tuple (S₁, …, S_d) where S₁, …, S_d are pairwise disjoint subsets of S. If we have a sequence of d-compositions of a finite set and observe certain intersection patterns among parts of different compositions, what are the corresponding arithmetic constraints on the parameters of this sequence? When d = 1, many results in extremal combinatorics address this question. Bollobás set pair inequality is such a classic result for d = 2. In this note, we provide several arithmetic constraints for general d and propose a conjecture as a linear space analogue for one of them. Our study highlights the connection between extremal combinatorics and Young’s lattice of a rectangle
Naklonjenost vzgojiteljev in ostalih strokovnih delavcev v VIZ do meditacije in tehnik za umirjanje
No full textGeometrijske konstrukcije majhnih regularnihgrafov ožine 7
No full textWe present simple, geometric constructions for small regular graphs of girth 7 from the incidence graphs of some generalized quadrangles. We obtain infinite families of (q − 1)-regular, q-regular and (q + 1)-regular graphs of girth 7, for q a prime power. Some of them have the smallest order known so far
