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The B-Domatic Number of a Graph

Odile Favaron (2013)

Discussiones Mathematicae Graph Theory

Besides the classical chromatic and achromatic numbers of a graph related to minimum or minimal vertex partitions into independent sets, the b-chromatic number was introduced in 1998 thanks to an alternative definition of the minimality of such partitions. When independent sets are replaced by dominating sets, the parameters corresponding to the chromatic and achromatic numbers are the domatic and adomatic numbers d(G) and ad(G). We introduce the b-domatic number bd(G) as the counterpart of the...

The bondage number of graphs: good and bad vertices

Vladimir Samodivkin (2008)

Discussiones Mathematicae Graph Theory

The domination number γ(G) of a graph G is the minimum number of vertices in a set D such that every vertex of the graph is either in D or is adjacent to a member of D. Any dominating set D of a graph G with |D| = γ(G) is called a γ-set of G. A vertex x of a graph G is called: (i) γ-good if x belongs to some γ-set and (ii) γ-bad if x belongs to no γ-set. The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination...

The box parameter for words and permutations

Helmut Prodinger (2014)

Open Mathematics

The box parameter for words counts how often two letters w j and w k define a “box” such that all the letters w j+1; ..., w k−1 fall into that box. It is related to the visibility parameter and other parameters on words. Three models are considered: Words over a finite alphabet, permutations, and words with letters following a geometric distribution. A typical result is: The average box parameter for words over an M letter alphabet is asymptotically given by 2n − 2n H M/M, for fixed M and n → ∞.

The Bruhat rank of a binary symmetric staircase pattern

Zhibin Du, Carlos M. da Fonseca (2016)

Open Mathematics

In this work we show that the Bruhat rank of a symmetric (0,1)-matrix of order n with a staircase pattern, total support, and containing In, is at most 2. Several other related questions are also discussed. Some illustrative examples are presented.

The Cayley graph and the growth of Steiner loops

P. Plaumann, L. Sabinina, I. Stuhl (2014)

Commentationes Mathematicae Universitatis Carolinae

We study properties of Steiner loops which are of fundamental importance to develop a combinatorial theory of loops along the lines given by Combinatorial Group Theory. In a summary we describe our findings.

The centre of a Steiner loop and the maxi-Pasch problem

Andrew R. Kozlik (2020)

Commentationes Mathematicae Universitatis Carolinae

A binary operation “ · ” which satisfies the identities x · e = x , x · x = e , ( x · y ) · x = y and x · y = y · x is called a Steiner loop. This paper revisits the proof of the necessary and sufficient conditions for the existence of a Steiner loop of order n with centre of order m and discusses the connection of this problem to the question of the maximum number of Pasch configurations which can occur in a Steiner triple system (STS) of a given order. An STS which attains this maximum for a given order is said to be maxi-Pasch. We show that...

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