TY - JOUR

T1 - Half-integral packing of odd cycles through prescribed vertices

AU - Kakimura, Naonori

AU - Kawarabayashi, Ken Ichi

N1 - Funding Information:
†Partly supported by Grant-in-Aid for Scientific Research and JST, ERATO, Kawarabayashi Large Graph Project.

PY - 2013/10

Y1 - 2013/10

N2 - The well-known theorem of Erdo{double acute}s-Pósa says that a graph G has either k disjoint cycles or a vertex set X of order at most f(k) for some function f such that G\X is a forest. Starting with this result, there are many results concerning packing and covering cycles in graph theory and combinatorial optimization. In this paper, we discuss packing disjoint S-cycles, i.e., cycles that are required to go through a set S of vertices. For this problem, Kakimura-Kawarabayashi-Marx (2011) and Pontecorvi-Wollan (2010) recently showed the Erdo{double acute}s-Pósa-type result holds. We further try to generalize this result to packing S-cycles of odd length. In contrast to packing S-cycles, the Erdo{double acute}s-Pósa-type result does not hold for packing odd S-cycles. We then relax packing odd S-cycles to half-integral packing, and show the Erdo{double acute}s-Pósa-type result for the half-integral packing of odd S-cycles, which is a generalization of Reed (1999) when S=V. That is, we show that given an integer k and a vertex set S, a graph G has either 2k odd S-cycles so that each vertex is in at most two of these cycles, or a vertex set X of order at most f(k) (for some function f) such that G\X has no odd S-cycle.

AB - The well-known theorem of Erdo{double acute}s-Pósa says that a graph G has either k disjoint cycles or a vertex set X of order at most f(k) for some function f such that G\X is a forest. Starting with this result, there are many results concerning packing and covering cycles in graph theory and combinatorial optimization. In this paper, we discuss packing disjoint S-cycles, i.e., cycles that are required to go through a set S of vertices. For this problem, Kakimura-Kawarabayashi-Marx (2011) and Pontecorvi-Wollan (2010) recently showed the Erdo{double acute}s-Pósa-type result holds. We further try to generalize this result to packing S-cycles of odd length. In contrast to packing S-cycles, the Erdo{double acute}s-Pósa-type result does not hold for packing odd S-cycles. We then relax packing odd S-cycles to half-integral packing, and show the Erdo{double acute}s-Pósa-type result for the half-integral packing of odd S-cycles, which is a generalization of Reed (1999) when S=V. That is, we show that given an integer k and a vertex set S, a graph G has either 2k odd S-cycles so that each vertex is in at most two of these cycles, or a vertex set X of order at most f(k) (for some function f) such that G\X has no odd S-cycle.

UR - http://www.scopus.com/inward/record.url?scp=84888408362&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84888408362&partnerID=8YFLogxK

U2 - 10.1007/s00493-013-2865-6

DO - 10.1007/s00493-013-2865-6

M3 - Article

AN - SCOPUS:84888408362

VL - 33

SP - 549

EP - 572

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 5

ER -