1996
DOI: 10.1002/(sici)1097-0118(199611)23:3<321::aid-jgt12>3.3.co;2-b
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Decomposing graphs under degree constraints

Abstract: We prove a conjecture of C. Thomassen: If s and t are non-negative integers, and if G is a graph with minimum degree s + t + 1, then the vertex set of G can be partitioned into two sets which induce subgraphs of minimum degree at least s and t , respectively.

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Cited by 50 publications
(56 citation statements)
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“…This has been used in the proofs of all parts of Theorem 16. As introduced by Stiebitz [66], a vertex partition (A, B) is associated with the weight…”
Section: Algorithm 1 Determination Of a Feasible Pair; Triangle-freementioning
confidence: 98%
“…This has been used in the proofs of all parts of Theorem 16. As introduced by Stiebitz [66], a vertex partition (A, B) is associated with the weight…”
Section: Algorithm 1 Determination Of a Feasible Pair; Triangle-freementioning
confidence: 98%
“…Thomassen [20] proved that the vertex set of a simple graph G (that is, a graph without multiple edges) with minimum degree at least 12d can be divided into two nonempty sets A, B such that the subgraphs G(A), G(B), induced by A, B, respectively, have minimum degree at least d. Subsequently, Stiebiz [15] proved the conjecture in [20] that the same conclusion holds if the minimum degree of G is at least 2d + 1 (and this is best possible).…”
Section: The Smallest Number Of Cycles In 3-connected Graphsmentioning
confidence: 99%
“…Proof: We delete successively vertices of degree at most q/p from G until we get a graph G , say, of minimum degree d > p/q = d/2. By the result in [15] the vertex set of G can be divided into sets A, B such that each of the graphs G (A), G (B) have minimum degree at least d /2 − 1 > d/4 − 1. Assume that A is no larger than B, that is, A has at most p/2 vertices.…”
Section: The Smallest Number Of Cycles In 3-connected Graphsmentioning
confidence: 99%
“…Motivated by this natural strategy, many work has been done along this line, and now we have a variety of results in this partition problem. To name a few, Stiebitz [8] showed a nice theorem, which states that every graph with minimum degree at least a + b + 1 can be decomposed into two parts A and B such that A has minimum degree at least a and B has minimum degree at least b. We see that the bound a + b + 1 is best possible by considering the complete graph of order a + b + 1.…”
Section: Introductionmentioning
confidence: 99%