TY - JOUR

T1 - Discrepancies of spanning trees and Hamilton cycles

AU - Gishboliner, Lior

AU - Krivelevich, Michael

AU - Michaeli, Peleg

N1 - Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2022/5

Y1 - 2022/5

N2 - We study the multicolour discrepancy of spanning trees and Hamilton cycles in graphs. As our main result, we show that under very mild conditions, the r-colour spanning-tree discrepancy of a graph G is equal, up to a constant, to the minimum s such that G can be separated into r equal parts by deleting s vertices. This result arguably resolves the question of estimating the spanning-tree discrepancy in essentially all graphs of interest. In particular, it allows us to immediately deduce as corollaries most of the results that appear in a recent paper of Balogh, Csaba, Jing and Pluhár, proving them in wider generality and for any number of colours. We also obtain several new results, such as determining the spanning-tree discrepancy of the hypercube. For the special case of graphs possessing certain expansion properties, we obtain exact asymptotic bounds. We also study the multicolour discrepancy of Hamilton cycles in graphs of large minimum degree, showing that in any r-colouring of the edges of a graph with n vertices and minimum degree at least [Formula presented], there must exist a Hamilton cycle with at least [Formula presented] edges in some colour. This extends a result of Balogh et al., who established the case r=2. The constant [Formula presented] in this result is optimal; it cannot be replaced by any smaller constant.

AB - We study the multicolour discrepancy of spanning trees and Hamilton cycles in graphs. As our main result, we show that under very mild conditions, the r-colour spanning-tree discrepancy of a graph G is equal, up to a constant, to the minimum s such that G can be separated into r equal parts by deleting s vertices. This result arguably resolves the question of estimating the spanning-tree discrepancy in essentially all graphs of interest. In particular, it allows us to immediately deduce as corollaries most of the results that appear in a recent paper of Balogh, Csaba, Jing and Pluhár, proving them in wider generality and for any number of colours. We also obtain several new results, such as determining the spanning-tree discrepancy of the hypercube. For the special case of graphs possessing certain expansion properties, we obtain exact asymptotic bounds. We also study the multicolour discrepancy of Hamilton cycles in graphs of large minimum degree, showing that in any r-colouring of the edges of a graph with n vertices and minimum degree at least [Formula presented], there must exist a Hamilton cycle with at least [Formula presented] edges in some colour. This extends a result of Balogh et al., who established the case r=2. The constant [Formula presented] in this result is optimal; it cannot be replaced by any smaller constant.

KW - Discrepancy

KW - Hamilton cycle

KW - Spanning tree

UR - http://www.scopus.com/inward/record.url?scp=85123583523&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2022.01.003

DO - 10.1016/j.jctb.2022.01.003

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AN - SCOPUS:85123583523

SN - 0095-8956

VL - 154

SP - 262

EP - 291

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

ER -