41,930 research outputs found
Metric Dimension: from Graphs to Oriented Graphs
International audienceThe metric dimension of an undirected graph is the cardinality of a smallest set of vertices that allows, through their distances to all vertices, to distinguish any two vertices of . Many aspects of this notion have been investigated since its introduction in the 70's, including its generalization to digraphs. In this work, we study, for particular graph families, the maximum metric dimension over all strongly-connected orientations, by exhibiting lower and upper bounds on this value. We first exhibit general bounds for graphs with bounded maximum degree. In particular, we prove that, in the case of subcubic -node graphs, all strongly-connected orientations asymptotically have metric dimension at most , and that there are such orientations having metric dimension . We then consider strongly-connected orientations of grids. For a torus with rows and columns, we show that the maximum value of the metric dimension of a strongly-connected Eulerian orientation is asymptotically (the equality holding when , are even, which is best possible). For a grid with rows and columns, we prove that all strongly-connected orientations asymptotically have metric dimension at most , and that there are such orientations having metric dimension
Metric Dimension: from Graphs to Oriented Graphs
International audienceThe metric dimension of an undirected graph is the cardinality of a smallest set of vertices that allows, through their distances to all vertices, to distinguish any two vertices of . Many aspects of this notion have been investigated since its introduction in the 70's, including its generalization to digraphs. In this work, we study, for particular graph families, the maximum metric dimension over all strongly-connected orientations, by exhibiting lower and upper bounds on this value. We first exhibit general bounds for graphs with bounded maximum degree. In particular, we prove that, in the case of subcubic -node graphs, all strongly-connected orientations asymptotically have metric dimension at most , and that there are such orientations having metric dimension . We then consider strongly-connected orientations of grids. For a torus with rows and columns, we show that the maximum value of the metric dimension of a strongly-connected Eulerian orientation is asymptotically (the equality holding when , are even, which is best possible). For a grid with rows and columns, we prove that all strongly-connected orientations asymptotically have metric dimension at most , and that there are such orientations having metric dimension
Quest for graphs of Frank number
In an orientation of the graph , the edge is deletable if and only
if is strongly connected. For a -edge-connected graph , H\"orsch
and Szigeti defined the Frank number as the minimum for which admits
orientations such that every edge of is deletable in at least one
of the orientations. They conjectured the Frank number is at most for
every -edge-connected graph . They proved the Petersen graph has Frank
number , but this was the only example with this property. We show an
infinite class of graphs having Frank number . H\"orsch and Szigeti showed
every -edge-colorable -edge-connected graph has Frank number at most .
It is tempting to consider non--edge-colorable graphs as candidates for
having Frank number greater than . Snarks are sometimes a good source of
finding critical examples or counterexamples. One might suspect various snarks
should have Frank number . However, we prove several candidate infinite
classes of snarks have Frank number . As well as the generalized Petersen
Graphs . We formulate numerous conjectures inspired by our
experience
Graphs with many strong orientations
We establish mild conditions under which a possibly irregular, sparse graph
has "many" strong orientations. Given a graph on vertices, orient
each edge in either direction with probability independently. We show
that if satisfies a minimum degree condition of and has
Cheeger constant at least , then the
resulting randomly oriented directed graph is strongly connected with high
probability. This Cheeger constant bound can be replaced by an analogous
spectral condition via the Cheeger inequality. Additionally, we provide an
explicit construction to show our minimum degree condition is tight while the
Cheeger constant bound is tight up to a factor.Comment: 14 pages, 4 figures; revised version includes more background and
minor changes that better clarify the expositio
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