2,239 research outputs found
On some third parts of nearly complete digraphs
AbstractFor the complete digraph DKn with n⩾3, its half as well as its third (or near-third) part, both non-self-converse, are exhibited. A backtracking method for generating a tth part of a digraph is sketched. It is proved that some self-converse digraphs are not among the near-third parts of the complete digraph DK5 in four of the six possible cases. For n=9+6k,k=0,1,…, a third part D of DKn is found such that D is a self-converse oriented graph and all D-decompositions of DKn have trivial automorphism group
A semi-exact degree condition for Hamilton cycles in digraphs
The paper is concerned with directed versions of Posa's theorem and Chvatal's
theorem on Hamilton cycles in graphs.
We show that for each a>0, every digraph G of sufficiently large order n
whose outdegree and indegree sequences d_1^+ \leq ... \leq d_n^+ and d_1^- \leq
>... \leq d_n^- satisfy d_i^+, d_i^- \geq min{i + a n, n/2} is Hamiltonian. In
fact, we can weaken these assumptions to
(i) d_i^+ \geq min{i + a n, n/2} or d^-_{n - i - a n} \geq n-i; (ii) d_i^-
\geq min{i + a n, n/2} or d^+_{n - i - a n} \geq n-i; and still deduce that G
is Hamiltonian. This provides an approximate version of a conjecture of
Nash-Williams from 1975 and improves a previous result of K\"uhn, Osthus and
Treglown
On the notion of balance in social network analysis
The notion of "balance" is fundamental for sociologists who study social
networks. In formal mathematical terms, it concerns the distribution of triad
configurations in actual networks compared to random networks of the same edge
density. On reading Charles Kadushin's recent book "Understanding Social
Networks", we were struck by the amount of confusion in the presentation of
this concept in the early sections of the book. This confusion seems to lie
behind his flawed analysis of a classical empirical data set, namely the karate
club graph of Zachary. Our goal here is twofold. Firstly, we present the notion
of balance in terms which are logically consistent, but also consistent with
the way sociologists use the term. The main message is that the notion can only
be meaningfully applied to undirected graphs. Secondly, we correct the analysis
of triads in the karate club graph. This results in the interesting observation
that the graph is, in a precise sense, quite "unbalanced". We show that this
lack of balance is characteristic of a wide class of starlike-graphs, and
discuss possible sociological interpretations of this fact, which may be useful
in many other situations.Comment: Version 2: 23 pages, 4 figures. An extra section has been added
towards the end, to help clarify some things. Some other minor change
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