Nonexistence of certain edge-girth-regular graphs
Abstract
Edge-girth-regular graphs (abbreviated as egr graphs) are regular graphs in which every edge is contained in the same number of shortest cycles. We prove that there is no 3-regular egr graph with girth 7 such that every edge is on exactly 6 shortest cycles, and there is no 3-regular egr graph with girth 8 such that every edge is on exactly 14 shortest cycles. This was conjectured by Goedgebeur and Jooken. A few other unresolved cases are settled as well.
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