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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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##### References:
 [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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