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The project Mali Savinjčani beremo as a means of promoting family reading during the preschool period
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Discovering and working with gifted and talented students in sports by Elementary Education students
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Kocke simetričnih dizajnov
No full textWe study n-dimensional matrices with {0, 1}-entries (n-cubes) such that all their 2-dimensional slices are incidence matrices of symmetric designs. A known construction of these objects obtained from difference sets is generalized so that the resulting n-cubes may have inequivalent slices. For suitable parameters, they can be transformed into n-dimensional Hadamard matrices with this property. In contrast, previously known constructions of n-dimensional designs all give examples with equivalent slices.Proučujemo n-dimenzionalne matrike z elementi iz množice {0, 1} (n-kocke) z lastnostjo, da so vsi njihovi 2-dimenzionalni prerezi incidenčne matrike simetričnih dizajnov. Znano konstrukcijo teh objektov na podlagi diferenčnih množic posplošimo tako, da imajo dobljene n-kocke lahko neekvivalentne prereze. Če uporabimo primerne parametre, jih lahko pretvorimo v n-dimenzionalne Hadamardove matrike s to lastnostjo. V nasprotju s to našo posplošeno konstukcijo vse doslej znane konstrukcije porodijo n-dimenzionalne dizajne z ekvivalentnimi prerezi
2-mavrično dominacijsko število kartezičnega produkta ciklov
No full textA k-rainbow dominating function (kRDF) of G is a function that assigns subsets of {1, 2, ..., k} to the vertices of G such that for vertices v with f(v) = ∅ we have
⋃{u ∈ N(v)}f(u) = {1, 2, ..., k}. The weight w(f) of a kRDF f is defined as
w(f) = ∑{v ∈ V(G)}|f(v)|. The minimum weight of a kRDF of G is called the k-rainbow domination number of G, which is denoted by γrk(G). In this paper, we study the 2-rainbow domination number of the Cartesian product of two cycles. Exact values are given for a number of infinite families and we prove lower and upper bounds for all other cases
Določanje zgornje meje za s pri točkovno primitivnih s-ločno tranzitivnih digrafih alternirajočih in simetričnih grup
No full textDetermining an upper bound on s for finite vertex-primitive s-arc-transitive digraphs has received considerable attention dating back to a question of Praeger in 1990. It was shown by Giudici and Xia that the smallest upper bound on s is attained for some digraph admitting an almost simple s-arc-transitive group. In this paper, based on the work of Pan, Wu and Yin, we prove that s<=2 in the case where the group is an alternating or symmetric group
Brooksovi tipi izrekov za barvne parametre lokalno končnih grafov in Kőnigova lema
No full textIn the past, analogues to Brooks’ theorem have been found for various parameters of graph coloring for infinite locally finite connected graphs in ZFC. We prove that there is a model of ZF (i.e., the Zermelo–Fraenkel set theory without the Axiom of Choice (AC)) where these theorems fail. Moreover, such theorems follow from Kőnig’s Lemma (every infinite locally finite connected graph has a ray–a weak form of AC) in ZF. In ZF, inspired by a combinatorial argument of Herrlich and Tachtsis from 2006, we formulate new conditions for the existence of the distinguishing chromatic number, the distinguishing chromatic index, the total chromatic number, the total distinguishing chromatic number, the odd chromatic number, and the neighbor-distinguishing index in infinite locally finite connected graphs, which are equivalent to Kőnig’s Lemma.
In this direction, we strengthen a recent result of Stawiski from 2023. We also generalize an algorithm of Imrich, Kalinowski, Pilśniak, and Shekarriz to show that the statement “If G is a connected infinite graph where the maximum degree Δ(G) ≥ 3 is finite, then the list-distinguishing chromatic number is at most 2Δ(G) − 1” holds under Kőnig’s Lemma in ZF. However, we prove that there is a model of ZF where the above statement fails
Horadamove kocke
No full textWe define and investigate a new three-parameter family of graphs that further generalizes the Fibonacci and metallic cubes. Namely, the number of vertices in this family of graphs satisfies Horadam recurrence, a linear recurrence of second order with constant coefficients. It is shown that the new family preserves many appealing and useful properties of the Fibonacci and metallic cubes. In particular, we present recursive decomposition and decomposition into grids and explore some metric and enumerative properties such as the number of edges, distribution of degrees, and cube polynomials. We also investigate the existence of Hamiltonian paths and cycles