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Degree-Continuous Graphs
Authors:John Gimbel  Ping Zhang
Institution:(1) University of Alaska, Fairbanks, AK, 99775;(2) Western Michigan University, Kalamazoo, MI, 49008-5152, U.S.A
Abstract:A graph G is degree-continuous if the degrees of every two adjacent vertices of G differ by at most 1. A finite nonempty set S of integers is convex if k isin S for every integer k with min(S)lesklesmax(S). It is shown that for all integers r > 0 and s ges0 and a convex set S with min(S) = r and max(S) = r+s, there exists a connected degree-continuous graph G with the degree set S and diameter 2s+2. The minimum order of a degree-continuous graph with a prescribed degree set is studied. Furthermore, it is shown that for every graph G and convex set S of positive integers containing the integer 2, there exists a connected degree-continuous graph H with the degree set S and containing G as an induced subgraph if and only if max(S)gesDelta(G) and G contains no r-regular component where r = max(S).
Keywords:distance  degree-continuous
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