## Abstract

For distinct vertices *u* and *v* in a graph *G*, the *connectivity* between *u* and *v*, denoted \(\kappa _G(u,v)\), is the maximum number of internally disjoint *u*–*v* paths in *G*. The *average connectivity* of *G*, denoted \({\overline{\kappa }}(G),\) is the average of \(\kappa _G(u,v)\) taken over all unordered pairs of distinct vertices *u*, *v* of *G*. Analogously, for a directed graph *D*, the *connectivity* from *u* to *v*, denoted \(\kappa _D(u,v)\), is the maximum number of internally disjoint directed *u*–*v* paths in *D*. The *average connectivity* of *D*, denoted \({\overline{\kappa }}(D)\), is the average of \(\kappa _D(u,v)\) taken over all ordered pairs of distinct vertices *u*, *v* of *D*. An *orientation* of a graph *G* is a directed graph obtained by assigning a direction to every edge of *G*. For a graph *G*, let \({\overline{\kappa }}_{\max }(G)\) denote the maximum average connectivity among all orientations of *G*. In this paper we obtain bounds for \({\overline{\kappa }}_{\max }(G)\) and for the ratio \({\overline{\kappa }}_{\max }(G)/{\overline{\kappa }}(G)\) for all graphs *G* of a given order and in a given class of graphs. Whenever possible, we demonstrate sharpness of these bounds. This problem had previously been studied for trees. We focus on the classes of cubic 3-connected graphs, minimally 2-connected graphs, 2-trees, and maximal outerplanar graphs.

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## Notes

The family of trees described in Henning and Oellermann (2004) for which the lower bound is asymptotically sharp can be obtained as follows. For a given \(t \ge 1\), take three copies of \(K_{1,t}\), and identify a leaf from each copy of \(K_{1,t}\) in a single vertex (which will have degree 3). Let \(T_{3t+1}\) be such a tree. So if \(n=3t+1\), then \(\bar{\kappa }_{\max }(T_{3t+1}) = \frac{2n^2+14n-43}{9n(n-1)}\). We point out that it was incorrectly stated in Henning and Oellermann (2004) that for a tree

*T*of order \(n \ge 3\), we have \(\bar{\kappa }_{\max }(T) \ge \frac{2n^2+14n-43}{9n(n-1)}\). This inequality holds for \(n \ge 34\). However, for \(n<34\), we have$$\begin{aligned} \bar{\kappa }_{\max }(K_{1,n-1})&=\frac{\left\lfloor \frac{n-1}{2}\right\rfloor \left\lceil \frac{n-1}{2}\right\rceil +(n-1)}{n(n-1)}< \frac{2n^2+14n-43}{9n(n-1)}. \end{aligned}$$For \(n< 34\), the stars are in fact the extremal trees. That is, for \(n<34\), one can show that \(\bar{\kappa }_{\max }(K_{1,n-1}) \le \bar{\kappa }_{\max }(T)\) for every tree

*T*of order*n*.

## References

Beineke LW, Oellermann OR, Pippert RE (2002) The average connectivity of a graph. Discrete Math 252:31–45

Bollobás B (2004) Extremal graph theory. Dover Publications, Mineola

Casablanca RM, Mol L, Oellermann OR (2021) Average connectivity of minimally \(2\)-connected graphs and average edge-connectivity of minimally \(2\)-edge-connected graphs. Discrete Appl Math 289:233–247

Chvátal V (1973) Tough graphs and Hamiltonian circuits. Discrete Math 5:215–228

Dankelmann P, Oellermann OR (2003) On the average connectivity of a graph. Discrete Appl Math 129:305–318

Dirac GA (1967) Minimally \(2\)-connected graphs. J Reine Angew Math 228:204–216

Durand de Gevigney O (2020) On Frank’s conjecture on \(k\)-connected orientations. J Combin Theory Ser B 141:105–114

Henning MA, Oellermann OR (2004) The average connectivity of a digraph. Discrete Appl Math 140:143–153

Henning MA, Oellermann OR (2001) The average connectivity of regular multipartite tournaments. Australas J Combin 23:101–113

Mader W (1972) Ecken vom Grad \(n\) in minimalen \(n\)-fach zusammenhängenden Graphen. Arch Math 23:219–224

Mader W (1978) A reduction method for edge-connectivity in graphs. Ann Discrete Math 3:145–164

Nash-Williams CSJA (1960) On orientations, connectivity and odd-vertex-pairings in finite graphs. Can J Math 12:555–567

Oellermann OR (2013) Menger’s theorem. In: Beineke LW, Wilson RJ (eds) Topics in structural graph theory. Cambridge University Press, Cambridge

Plummer MD (1968) On minimal blocks. Trans. Am. Math. Soc. 134(1):85–94

Robbins HE (1939) Questions, discussions, and notes: a theorem on graphs, with an application to a problem of traffic control. Am. Math. Mon. 46:281–283

Thomassen C (1989) Configurations in graphs of large minimum degree, connectivity, or chromatic number. In: Proceedings of the third international conference on combinatorial mathematics, New York 1985, Annals of the New York Academy of Sciences, New York, vol 555, pp 402–412

Thomassen C (2014) Strongly \(2\)-connected orientations of graphs. J Combin Theory Ser B 110:67–78

## Acknowledgements

The authors wish to thank the Banff International Research Station for their support of the focussed research group number 18frg233, “Measuring the Connectedness of Graphs and Digraphs”, during which most of the research for this manuscript was completed.

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Ortrud Oellermann supported by an NSERC Grant CANADA, Grant number RGPIN-2016-05237. Lucas Mol supported by an NSERC Grant CANADA, Grant number RGPIN-2021-04084.

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Casablanca, R.M., Dankelmann, P., Goddard, W. *et al.* The maximum average connectivity among all orientations of a graph.
*J Comb Optim* **43**, 543–570 (2022). https://doi.org/10.1007/s10878-021-00789-z

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DOI: https://doi.org/10.1007/s10878-021-00789-z