Partitionable graphs arising from near-factorizations of finite groups

AbstractIn 1979, two constructions for making partitionable graphs were introduced in (by Chvátal et al. (Ann. Discrete Math. 21 (1984) 197)). The graphs produced by the second construction are called CGPW graphs. A near-factorization (A,B) of a finite group is roughly speaking a non-trivial factorization of G minus one element into two subsets A and B. Every CGPW graph with n vertices turns out to be a Cayley graph of the cyclic group Zn, with connection set (A−A)⧹{0}, for a near-factorization (A,B) of Zn. Since a counter-example to the Strong Perfect Graph Conjecture would be a partitionable graph (Padberg, Math. Programming 6 (1974) 180), any ‘new’ construction for making partitionable graphs is of interest. In this paper, we investigate the near-factorizations of finite groups in general, and their associated Cayley graphs which are all partitionable. In particular, we show that near-factorizations of the dihedral groups produce every CGPW graph of even order. We present some results about near-factorizations of finite groups which imply that a finite abelian group with a near-factorization (A,B) such that |A|⩽4 must be cyclic (already proved by De Caen et al. (Ars Combin. 29 (1990) 53)). One of these results may be used to speed up exhaustive calculations. At last, we prove that there is no counter-example to the Strong Perfect Graph Conjecture arising from near-factorizations of a finite abelian group of even order

The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution

International audienceThe Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The first of these three approaches yielded the first (and to date only) proof of the SPGC; the other two remain promising to consider in attempting an alternative proof. This paper is an unbalanced survey of the attempts to solve the SPGC; unbalanced, because (1) we devote a signicant part of it to the 'primitive graphs and structural faults' paradigm which led to the Strong Perfect Graph Theorem (SPGT); (2) we briefly present the other "direct" attempts, that is, the ones for which results exist showing one (possible) way to the proof; (3) we ignore entirely the "indirect" approaches whose aim was to get more information about the properties and structure of perfect graphs, without a direct impact on the SPGC. Our aim in this paper is to trace the path that led to the proof of the SPGT as completely as possible. Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin and B.A. Reed, eds., Perfect Graphs (Wiley & Sons, 2001).], but it also implies a deeper analysis (with additional results) and another viewpoint on the topic

On topics related to sum systems

For m ∈ N, we say that the m integer sets A1, . . . , Am ⊂ N0, form an m-part sum system if their sumset is the target set Xm j=1 Aj = n a1 + · · · + am : aj ∈ Aj , j ∈ {1, . . . , m} o = { 0, 1, 2, . . . ,Ym j=1 |Aj | − 1 } . That is to say, the sum over each element of the sets A1, . . . , Am uniquely generates the consecutive integers from 0 to Qm j=1 |Aj | − 1 with each integer appearing exactly once. Huxley, Lettington and Schmidt, in 2018, established a bijection between sum systems and sum-and-distance systems, utilising joint ordered factorisations, a specific form of ordered multi-factorisations, historically considered by MacMahon. They proved that for each m-part sum system there exists a corresponding m-part sum-and-distance system which generates the centro-symmetric set of consecutive (half) integers symmetric around the origin { − 1/2 (Ym j=1 |Aj | − 1 ), . . . , 1/2 (Ym j=1 |Aj | − 1 ) . In this thesis, we extend the results of Huxley, Lettington and Schmidt to obtain a unifying theory underpinning sum-and-distance systems, expressing their structures in terms of joint ordered-factorisations, thus enabling explicit construction formulae to be established via these factorisations. This unifying theory occurs when one allows consecutive half integers in the target set, when at least one component sum-and-distance set has even cardinality, leading to an invariance in the sum over weighted averages of the sum of squares across the sum-and-distance system component sets to be deduced. Further results include the application of associated divisor functions and Stirling numbers of the second kind, to enumerate all m-part joint ordered factorisations Nm(N) for a given positive integer N = n1 × n2 × . . . nm. We go on to show that the counting function Nm(N) satisfies an implicit three term recurrence relation proving an important relation in additive combinatorics. Additionally, sum systems (mod N + z), are considered, as well as orbit structures arising from very simple joint ordered factorisations. The latter leads to connections with cyclotomy

On Circular Critical Graphs

A graph on n vertices is called circular if its automorphism group contains an n-cycle. Let ω(G) and α(G) be, respectively, the clique number and the independence number of the graph G. A graph G with n vertices is called an (α, ω)-graph if 1. (1) n=α(G)ω(G)+1 2. (2) every vertex is in exactly α(G) maximum independent sets and α(G) maximum cliques, and 3. (3) each maximum clique intersects all but one maximum independent set, and vice versa. A graph is called critical if it is imperfect and all of its proper induced subgraphs are perfect. Lovasz and Padberg showed that all critical graphs are (α, ω)-graphs. Only one method is known for constructing circular (α, ω)-graphs. We show that the only critical graphs which arise from this construction are the odd, chordless cycles of length at least 5, and their complements