TY - JOUR
T1 - Graph decompositions without isolates
AU - Linial, Nathan
PY - 1984/2
Y1 - 1984/2
N2 - A. Frank (Problem session of the Fifth British Combinatorial Conference, Aberdeen, Scotland, 1975) conjectured that if G = (V, E) is a connected graph with all valencies ≥k and a1,...,ak ≥ 2 are integers with Σ ai = |V |, then V may be decomposed into subsets A1,...,Ak so that |Ai | = ai and the subgraph spanned by Ai in G has no isolated vertices (i = 1,...,k). The case k = 2 is proved in Maurer (J. Combin. Theory Ser. B 27 (1979), 294-319) along with some extensions. The conjecture for k = 3 and a result stronger than Maurer's extension for k = 2 are proved. A related characterization of a k-connected graph is also included in the paper, and a proof of the conjecture for the case a1 = a2 = ... = ak-1 = 2.
AB - A. Frank (Problem session of the Fifth British Combinatorial Conference, Aberdeen, Scotland, 1975) conjectured that if G = (V, E) is a connected graph with all valencies ≥k and a1,...,ak ≥ 2 are integers with Σ ai = |V |, then V may be decomposed into subsets A1,...,Ak so that |Ai | = ai and the subgraph spanned by Ai in G has no isolated vertices (i = 1,...,k). The case k = 2 is proved in Maurer (J. Combin. Theory Ser. B 27 (1979), 294-319) along with some extensions. The conjecture for k = 3 and a result stronger than Maurer's extension for k = 2 are proved. A related characterization of a k-connected graph is also included in the paper, and a proof of the conjecture for the case a1 = a2 = ... = ak-1 = 2.
UR - http://www.scopus.com/inward/record.url?scp=48749140411&partnerID=8YFLogxK
U2 - 10.1016/0095-8956(84)90010-8
DO - 10.1016/0095-8956(84)90010-8
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AN - SCOPUS:48749140411
SN - 0095-8956
VL - 36
SP - 16
EP - 25
JO - Journal of Combinatorial Theory. Series B
JF - Journal of Combinatorial Theory. Series B
IS - 1
ER -