Maximum diameter of regular digraphs

Soares, Jacyra Ramos

doi:10.1002/jgt.3190160505

Cited by 17 publications

(6 citation statements)

References 8 publications

(6 reference statements)

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“…For further known results on the maximum diameter of regular (di)graphs, the interested reader is referred to [5], for example.…”

Section: Proof Of Propositionmentioning

confidence: 99%

A Cauchy–Davenport Type Result for Arbitrary Regular Graphs

Hegarty

2011

Integers

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Motivated by the Cauchy-Davenport theorem for sumsets, and its interpretation in terms of Cayley graphs, we prove the following main result: There is a universal constant > 0 such that, if G is a connected, regular graph on n vertices, then either every pair of vertices can be connected by a path of length at most three, or the number of pairs of such vertices is at least 1 C times the number of edges in G . We discuss a range of further questions to which this result gives rise.

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“…For further known results on the maximum diameter of regular (di)graphs, the interested reader is referred to [5], for example.…”

Section: Proof Of Propositionmentioning

confidence: 99%

A Cauchy–Davenport Type Result for Arbitrary Regular Graphs

Hegarty

2011

Integers

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“…As shown by Soares [3], the above bound does not hold for digraphs. He constructed graphs of order n, minimum degree δ (defined as the minimum over all in-degrees and all out-degrees) and diameter n − 2δ + 1.…”

Section: Introductionmentioning

confidence: 92%

The Diameter of Almost Eulerian Digraphs

Dankelmann

Volkmann

2010

Electron. J. Combin.

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Soares [J. Graph Theory 1992] showed that the well known upper bound $\frac{3}{\delta+1}n+O(1)$ on the diameter of undirected graphs of order $n$ and minimum degree $\delta$ also holds for digraphs, provided they are eulerian. In this paper we investigate if similar bounds can be given for digraphs that are, in some sense, close to being eulerian. In particular we show that a directed graph of order $n$ and minimum degree $\delta$ whose arc set can be partitioned into $s$ trails, where $s\leq \delta-2$, has diameter at most $3 ( \delta+1 - \frac{s}{3})^{-1}n+O(1)$. If $s$ also divides $\delta-2$, then we show the diameter to be at most $3(\delta+1 - \frac{(\delta-2)s}{3(\delta-2)+s} )^{-1}n+O(1)$. The latter bound is sharp, apart from an additive constant. As a corollary we obtain the sharp upper bound $3( \delta+1 - \frac{\delta-2}{3\delta-5})^{-1} n + O(1)$ on the diameter of digraphs that have an eulerian trail.

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“…If G is regular of degree r, its diameter cannot exceed 3n−r−3 r+1 [11]. Since the drunk will step toward the cop with probability at least 1/r at each move, resulting (after her response) in a decrease of 2 in their distance, the expected capture time is bounded by r · diam(G)/2 < 3n/2.…”

Section: Every Point Lies On Exactly N + 1 Distinct Linesmentioning

confidence: 99%

Capturing the Drunk Robber on a Graph

Komarov

Winkler

2014

Electron. J. Combin.

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We show that the expected time for a smart "cop" to catch a drunk "robber" on an n-vertex graph is at most n + o(n). More precisely, let G be a simple, connected, undirected graph with distinguished points u and v among its n vertices. A cop begins at u and a robber at v; they move alternately from vertex to adjacent vertex. The robber moves randomly, according to a simple random walk on G; the cop sees all and moves as she wishes, with the object of "capturing" the robber-that is, occupying the same vertex-in least expected time. We show that the cop succeeds in expected time no more than n + o(n). Since there are graphs in which capture time is at least n − o(n), this is roughly best possible. We note also that no function of the diameter can be a bound on capture time.

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Maximum diameter of regular digraphs

Cited by 17 publications

References 8 publications

A Cauchy–Davenport Type Result for Arbitrary Regular Graphs

A Cauchy–Davenport Type Result for Arbitrary Regular Graphs

The Diameter of Almost Eulerian Digraphs

Capturing the Drunk Robber on a Graph

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