2,684 research outputs found

### Line digraphs and coreflexive vertex sets

The concept of coreflexive set is introduced to study the structure of
digraphs. New characterizations of line digraphs and nth-order line digraphs
are given. Coreflexive sets also lead to another natural way of forming an
intersection digraph from a given digraph.Comment: 8 pages, 3 figure

### Extremal Values of the Interval Number of a Graph

The interval number $i( G )$ of a simple graph $G$ is the smallest number $t$ such that to each vertex in $G$ there can be assigned a collection of at most $t$ finite closed intervals on the real line so that there is an edge between vertices $v$ and $w$ in $G$ if and only if some interval for $v$ intersects some interval for $w$. The well known interval graphs are precisely those graphs $G$ with $i ( G )\leqq 1$. We prove here that for any graph $G$ with maximum degree $d, i ( G )\leqq \lceil \frac{1}{2} ( d + 1 ) \rceil$. This bound is attained by every regular graph of degree $d$ with no triangles, so is best possible. The degree bound is applied to show that $i ( G )\leqq \lceil \frac{1}{3}n \rceil$ for graphs on $n$ vertices and $i ( G )\leqq \lfloor \sqrt{e} \rfloor$ for graphs with $e$ edges

### To catch a falling robber

We consider a Cops-and-Robber game played on the subsets of an $n$-set. The
robber starts at the full set; the cops start at the empty set. On each turn,
the robber moves down one level by discarding an element, and each cop moves up
one level by gaining an element. The question is how many cops are needed to
ensure catching the robber when the robber reaches the middle level. Aaron Hill
posed the problem and provided a lower bound of $2^{n/2}$ for even $n$ and
$\binom{n}{\lceil n/2 \rceil}2^{-\lfloor n/2 \rfloor}$ for odd $n$. We prove an
upper bound (for all $n$) that is within a factor of $O(\ln n)$ times this
lower bound.Comment: Minor revision

### Star-factors of tournaments

Let S_m denote the m-vertex simple digraph formed by m-1 edges with a common
tail. Let f(m) denote the minimum n such that every n-vertex tournament has a
spanning subgraph consisting of n/m disjoint copies of S_m. We prove that m lg
m - m lg lg m <= f(m) <= 4m^2 - 6m for sufficiently large m.Comment: 5 pages, 1 figur

### Short Proofs for Cut-and-Paste Sorting of Permutations

We consider the problem of determining the maximum number of moves required
to sort a permutation of $[n]$ using cut-and-paste operations, in which a
segment is cut out and then pasted into the remaining string, possibly
reversed. We give short proofs that every permutation of $[n]$ can be
transformed to the identity in at most \flr{2n/3} such moves and that some
permutations require at least \flr{n/2} moves.Comment: 7 pages, 2 figure

- …