A note on total colourings of digraphs

In the paper two different arc-colourings and two associated with the total colourings of digraphs are considered. In one of these colourings we show that the problem of calculating the total chromatic index reduces to that of calculating the chromatic number of the underlying graph. In the other colouring we find the total chromatic indices of complete symmetric digraphs and tournaments.     

Research problems from the BCC22

Open problems from the problem session at the 22nd British Combinatorial Conference at St Andrews, on 10 July 2009.     

Modal Aggregation and the Theory of Paraconsistent Filters

This paper articulates the structure of a two species of weakly aggregative necessity in a common idiom, neighbourhood semantics, using the notion of a k-filter of propositions. A k-filter on a non-empty set I is a collection of subsets of I which (i) contains I, (ii) is closed under supersets on I, and (iii) contains ∪{X_i ≤ X_j : 0 ≤ i < j ≤ k} whenever it contains the subsets X₀,…, X_k. The mathematical content of the proof that weakly aggregative modal logic is complete relative to k-ary frame theory, the standard semantic idiom for weakly aggregative modal logic (see [1]) is presented in language-independent terms as a representation theorem for k-filters: every non-trivial k-filter is included in the union of ≤ k non-trivial filters. The elementary theory of k-filters is developed and then applied in the form of an ultrafilter extension result for k-ary frame theory. Mathematics Subject Classification: 03B45.     

Some colouring problems for Paley graphs

The Paley graph P_q, where is a prime power, is the graph with vertices the elements of the finite field F_q and an edge between x and y if and only if x-y is a non-zero square in F_q. This paper gives new results on some colouring problems for Paley graphs and related discussion.     

Coloured solutions of equations in finite groups

In this article, we consider the relations between colourings and some equations in finite groups. We will express relations linking the numbers of the differently coloured solutions of an equation that depend only on the cardinality of the colouring and not on the distribution of the colours. This gives a link between Ramsey theory that investigates the existence of monochromatic solutions and what is now called anti-Ramsey theory that investigates the existence of rainbow solutions. Both theories are in expansion. We will apply these results to the counting of rainbow 3-term arithmetic progressions in any abelian group with equinumerous three-colouring and to the counting of points on a conic defined on a finite field. We will end by discussing the generalised case of a system of equations.     

Topologically 4-chromatic graphs and signatures of odd cycles

We investigate group-theoretic “signatures” of odd cycles of a graph, and their connections to topological obstructions to 3-colourability. In the case of signatures derived from free groups, we prove that the existence of an odd cycle with trivial signature is equivalent to having the coindex of the hom-complex at least 2 (which implies that the chromatic number is at least 4). In the case of signatures derived from elementary abelian 2-groups we prove that the existence of an odd cycle with trivial signature is a sufficient condition for having the index of the hom-complex at least 2 (which again implies that the chromatic number is at least 4).     

A list version of Dirac's theorem on the number of edges in colour‐critical graphs

One of the basic results in graph colouring is Brooks' theorem [R. L. Brooks, Proc Cambridge Phil Soc 37 ( 4 ) 194–197], which asserts that the chromatic number of every connected graph, that is not a complete graph or an odd cycle, does not exceed its maximum degree. As an extension of this result, Dirac [G. A. Dirac, Proc London Math Soc 7(3) ( 7 ) 161–195] proved that every k‐colour‐critical graph (k ≥ 4) on n ≥ k + 2 vertices has at least ½((k ? 1) n + k ? 3) edges. The aim of this paper is to prove a list version of Dirac's result and to extend it to hypergraphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 165–177, 2002; DOI 10.1002/jgt.998     

Avoiding multiple entry arrays

In this paper we consider the problem of avoiding arrays with more than one entry per cell. An n × n array on n symbols is said to be if an n × n latin square, on the same symbols, can be found which differs from the array in every cell. Our first result is for chessboard squares with at most two entries per black cell. We show that if k ≥ 1 and C is a 4k × 4k chessboard square on symbols 1, 2, …, 4k in which every black cell contains at most two symbols and every symbol appears at most once in every row and column, then C is avoidable. Our main result is for squares with at most two entries in any cell and answers a question of Hilton. If k 3240 and F is a 4k × 4k array on 1, 2,…, 4k in which every cell contains at most two symbols and every symbol appears at most twice in every row and column, then F is avoidable. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 257–266, 1997     

On cycle systems with specified weak chromatic number

A weak k-colouring of an m-cycle system is a colouring of the vertices of the system with k colours in such a way that no cycle of the system has all of its vertices receive the same colour. An m-cycle system is said to be weakly k-chromatic if it has a weak k-colouring but no weak (k−1)-colouring. In this paper we show that for all k?2 and m?3 with (k,m)≠(2,3) there is a weakly k-chromatic m-cycle system of order v for all sufficiently large admissible v.