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Displaying similar documents to “Note on independent sets of a graph”

Degree-continuous graphs

John Gimbel, Ping Zhang (2001)

Czechoslovak Mathematical Journal

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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 ) ≤ k ≤ max ( 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 2 s + 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...

F -continuous graphs

Gary Chartrand, Elzbieta B. Jarrett, Farrokh Saba, Ebrahim Salehi, Ping Zhang (2001)

Czechoslovak Mathematical Journal

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For a nontrivial connected graph F , the F -degree of a vertex v in a graph G is the number of copies of F in G containing v . A graph G is F -continuous (or F -degree continuous) if the F -degrees of every two adjacent vertices of G differ by at most 1. All P 3 -continuous graphs are determined. It is observed that if G is a nontrivial connected graph that is F -continuous for all nontrivial connected graphs F , then either G is regular or G is a path. In the case of a 2-connected graph F , however,...

On integral sum graphs with a saturated vertex

Zhibo Chen (2010)

Czechoslovak Mathematical Journal

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As introduced by F. Harary in 1994, a graph G is said to be an i n t e g r a l s u m g r a p h if its vertices can be given a labeling f with distinct integers so that for any two distinct vertices u and v of G , u v is an edge of G if and only if f ( u ) + f ( v ) = f ( w ) for some vertex w in G . We prove that every integral sum graph with a saturated vertex, except the complete graph K 3 , has edge-chromatic number equal to its maximum degree. (A vertex of a graph G is said to be if it is adjacent to every...