Critical chromatic number and the complexity of perfect packings in graphs

Let <i>H</i> be any non-bipartite graph. We determine asymptotically the minimum degree of a graph <i>G</i> which ensures that <i>G</i> has a perfect <i>H</i>-packing. More precisely, we determine the smallest number τ having the following property: For every positive constant γ there exists an integer <i>n</i><inf>0</inf> = <i>n</i><inf>0</inf>(γ, <i>H</i>) such that every graph <i>G</i> whose order <i>n</i> ≥<i>n</i><inf>0</inf> is divisible by |<i>H</i>| and whose minimum degree is at least (τ + γ) <i>n</i> contains a perfect <i>H</i>-packing. The value of τ depends on the relative sizes of the colour classes in the optimal colourings of <i>H.</i> The proof is algorithmic, which shows that the problem of finding a maximum <i>H</i>-packing is polynomially solvable for graphs <i>G</i> whose minimum degree is at least (τ + γ)<i>n.</i> On the other hand, given any positive constant γ, we show that for infinitely many (non-bipartite) graphs <i>H</i> the corresponding decision problem becomes NP-complete if one considers input graphs <i>G</i> of minimum degree at least (τ - γ)<i>n.</i>

[1]  Ken-ichi Kawarabayashi.  K 4 - -factor in a graph , 2002 .

[2]  Brenda S. Baker,et al.  Approximation algorithms for NP-complete problems on planar graphs , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[3]  János Komlós,et al.  Blow-up Lemma , 1997, Combinatorics, Probability and Computing.

[4]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[5]  Alexander Schrijver,et al.  On the Size of Systems of Sets Every t of Which Have an SDR, with an Application to the Worst-Case Ratio of Heuristics for Packing Problems , 1989, SIAM J. Discret. Math..

[6]  Brenda S. Baker.  Approximation Algorithms for NP-Complete Problems on Planar Graphs (Preliminary Version) , 1983, FOCS.

[7]  János Komlós,et al.  Tiling Turán Theorems , 2000, Comb..

[8]  Daniela Kühn,et al.  Large planar subgraphs in dense graphs , 2005, J. Comb. Theory, Ser. B.

[9]  János Komlós,et al.  An algorithmic version of the blow-up lemma , 1998, Random Struct. Algorithms.

[10]  János Komlós,et al.  Proof of the Alon-Yuster conjecture , 2001, Discret. Math..

[11]  M. Simonovits,et al.  Szemeredi''s Regularity Lemma and its applications in graph theory , 1995 .

[12]  David G. Kirkpatrick,et al.  On the Complexity of General Graph Factor Problems , 1981, SIAM J. Comput..

[13]  Noga Alon,et al.  AlmostH-factors in dense graphs , 1992, Graphs Comb..

[14]  Francine Berman,et al.  Generalized Planar Matching , 1990, J. Algorithms.

[15]  Ali Shokoufandeh,et al.  Proof of a tiling conjecture of Komlós , 2003, Random Struct. Algorithms.

[16]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[17]  Gábor N. Sárközy,et al.  An algorithmic version of the blow-up lemma , 1998 .

[18]  Vojtech Rödl,et al.  The Algorithmic Aspects of the Regularity Lemma , 1994, J. Algorithms.

[19]  Viggo Kann,et al.  Maximum Bounded H-Matching is MAX SNP-Complete , 1994, Inf. Process. Lett..