Arithmetic progressions of cycles in outer-planar graphs

AbstractA question of Erdős asks if every graph with minimum degree 3 must contain a pair of cycles whose lengths differ by 1 or 2. Some recent work of Häggkvist and Scott (see Arithmetic progressions of cycles in graphs, preprint), whilst proving this, also shows that minimum degree 500k2 guarantees the existence of cycles whose lengths are m,m+2,m+4,…,m+2k for some m—an arithmetic progression of cycles. In like vein, we prove that an outer-planar graph of order n, with bounded internal face size, and outer face a cycle, must contain a sequence of cycles whose lengths form an arithmetic progression of length exp((clogn)1/3−loglogn). Using this we give an answer for outer-planar graphs to a question of Erdős concerning the number of different sets which can be achieved as cycle spectra

A Generalization of a Theorem of Dirac

AbstractIn this paper, we give a generalization of a well-known result of Dirac that given any k vertices in a k-connected graph where k⩾2, there is a circuit containing all of them. We also generalize a result of Häggkvist and Thomassen. Our main result partially answers an open matroid question of Oxley

Completing some Partial Latin Squares

AbstractWe show that any partial 3 r× 3 r Latin square whose filled cells lie in two disjoint r×r sub-squares can be completed. We do this by proving the more general result that any partial 3 r by 3 r Latin square, with filled cells in the top left 2r× 2 r square, for which there is a pairing of the columns so that in each row there is a filled cell in at most one of each matched pair of columns, can be completed if and only if there is some way to fill the cells of the top left 2 r× 2 r square

The Independence Number of Graphs With Large Odd Girth

. Let G be an r-regular graph of order n and independence number #(G).We show that if G has odd girth 2k +3then #(G) # n 1-1/k r 1/k . We also prove similar results for graphs which are not regular. Using these results we improve on the lower bound of Monien and Speckenmeyer, for the independence number of a graph of order n and odd girth 2k +3. AMS Subject Classification. 05C15 1. Introduction Let G be a triangle--free graph of order n with average degree d, and independence number #(G). There has been great interest in finding good lower bounds for #(G)intermsofd, and producing polynomial--time algorithms which find large independent sets of G. In [1] and [2] Ajtai, Komlos and Szemeredi made a breakthrough in this area when they provided a polynomial algorithm to find an independent set of size at least #(G) # n log d 100d . * Correspondence to Tristan Denley, Matematiska institutionen, Umeauniversitet,Umea, Sweden Email to [email protected] 1 ### ############..

On a generalization of the Evans Conjecture

The Evans Conjecture states that a partial Latin square of order n with at most n − 1 entries can be completed. In this paper we generalize the Evans Conjecture by showing that a partial r-multi Latin square of order n with at most n−1 entries can be completed. Using this generalization, we confirm a case of a conjecture of Häggkvist.Journal ArticlePublishe