Connected graphs of genus g with complementary orbits

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Abstract

This paper concerns graphs with exactly two orbits under the action of their automorphism group; let their sizes be p and q respectively where p≤q, p=p′d, q=q′d, and d=gcd(p,q). It is shown that there is a finite set Fg of pairs of natural numbers such that if G is a connected graph of genus g with complementary orbits of sizes p and q, then either p′≤2 or (p, q)ε{lunate}Fg For connected planar and toroidal graphs the sizes of all complementary orbits are determined precisely, and the graphs which give rise to elements of F0 and F1 are characterized. © 1983.

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Hutchinson, J. P., & McNulty, G. F. (1983). Connected graphs of genus g with complementary orbits. Discrete Mathematics, 45(2–3), 255–275. https://doi.org/10.1016/0012-365X(83)90042-0

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