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- W2044686427 abstract "Let X = {x1,…,xm} and Y = {y1,…,ym} be two disjoints sets of vertices in a graph G. Then (X,Y) is called an antipodal set-pair of size m(m-ASP, for short) if the distance of xi and yj is at most two if and only if i ≠ j. We prove that in a graph of maximum degree k every m-ASP has size m⩽k(k + 1)2 + 1. This upper bound is nearly best possible since, for every k⩾2, there exists a regular graph of degree k, with an m-ASP, m⩾k(k+1)2 (and m = k(k + 1)2 + 1 when k ≡ 0 or 1 (mod 4)). If the degrees of the xi and yj are bounded above by p and q, respectively, then an m-ASP can exist only for m⩽(p + q + 1p). We conjecture that this bound cán be improved to m⩽(p + pq), and verify this conjecture when the graph satisfies some additional assumptions." @default.
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- W2044686427 date "1993-02-01" @default.
- W2044686427 modified "2023-09-23" @default.
- W2044686427 title "Bounded degrees and prescribed distances in graphs" @default.
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- W2044686427 doi "https://doi.org/10.1016/0012-365x(93)90144-i" @default.
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