Prime and polynomial distances in colourings of the plane

We give two extensions of the recent theorem of the first author that the odd distance graph has unbounded chromatic number. The first is that for any non-constant polynomial $f$ with integer coefficients and positive leading coefficient, every finite colouring of the plane contains a monochromatic pair of distinct points whose distance is equal to $f(n)$ for some integer $n$. The second is that for every finite colouring of the plane, there is a monochromatic pair of points whose distance is a prime number.Comment: 22 page

Distance-regular Cayley graphs with small valency

We consider the problem of which distance-regular graphs with small valency are Cayley graphs. We determine the distance-regular Cayley graphs with valency at most $4$, the Cayley graphs among the distance-regular graphs with known putative intersection arrays for valency $5$, and the Cayley graphs among all distance-regular graphs with girth $3$ and valency $6$ or $7$. We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs. We show that the incidence graphs of the known generalized hexagons are not Cayley graphs, and neither are some other distance-regular graphs that come from small generalized quadrangles or hexagons. Among some ``exceptional'' distance-regular graphs with small valency, we find that the Armanios-Wells graph and the Klein graph are Cayley graphs.Comment: 19 pages, 4 table

A family of mixed graphs with large order and diameter 2

A mixed regular graph is a connected simple graph in which each vertex has both a fixed outdegree (the same indegree) and a fixed undirected degree. A mixed regular graphs is said to be optimal if there is not a mixed regular graph with the same parameters and bigger order. We present a construction that provides mixed graphs of undirected degree qq, directed degree View the MathML sourceq-12 and order 2q22q2, for qq being an odd prime power. Since the Moore bound for a mixed graph with these parameters is equal to View the MathML source9q2-4q+34 the defect of these mixed graphs is View the MathML source(q-22)2-14. In particular we obtain a known mixed Moore graph of order 1818, undirected degree 33 and directed degree 11 called Bosák’s graph and a new mixed graph of order 5050, undirected degree 55 and directed degree 22, which is proved to be optimal.Peer ReviewedPostprint (author's final draft

QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time 2^{O~(n^{2/3})} for Maximum Clique on disk graphs. In stark contrast, Maximum Clique on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant ratio of approximation which cannot be attained even in time 2^{n^{1-epsilon}}, unless the Exponential Time Hypothesis fails