Degree-Continuous Graphs |
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Authors: | John Gimbel Ping Zhang |
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Institution: | (1) University of Alaska, Fairbanks, AK, 99775;(2) Western Michigan University, Kalamazoo, MI, 49008-5152, U.S.A |
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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 S for every integer k with min(S)kmax(S). It is shown that for all integers r > 0 and s 0 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)(G) and G contains no r-regular component where r = max(S). |
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Keywords: | distance degree-continuous |
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