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Interval-regularity does not lead to interval monotonicity. (English) Zbl 0784.05040
A connected graph \(G\) is interval-regular if for any two vertices \(u\) and \(v\) of \(G\) the number of neighbors of \(u\) lying on some shortest \((u,v)\)- path is equal to \(d(u,v)\). On the other hand \(G\) is interval-monotone if for any \(u\), \(v\) the set \(I(x,y)\) of vertices lying on shortest \((x,y)\)- paths is contained in the set of vertices \(I(u,v)\) on shortest \((u,v)\)- paths for \(x,y\in I(u,v)\). A 1980 conjecture of M. Mulder is that an interval-regular graph is interval-monotone. The conjecture is false: Presented is an infinite family of counterexamples reminiscent of the hypercubes \(Q_ n\).

MSC:
05C38 Paths and cycles
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[1] Foldes, S., A characterization of hypercubes, Discrete math., 17, 155-159, (1977) · Zbl 0354.05045
[2] Mulder, M., The interval function of a graph, () · Zbl 0446.05039
[3] Mulder, M., Interval-regular graphs, Discrete math., 41, 253-269, (1982) · Zbl 0542.05051
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