On cycle systems with specified weak chromatic number

Abstract

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.

On balanced incomplete block designs with specified weak chromatic number

“…The reason for these particular congruence conditions will become apparent when we employ these examples in Section 5 to establish the asymptotic existence of c-chromatic BIBDs for each c ≥ 2. Our approach in this section is inspired by a technique used in [4], and also bears similarities to methods used in [12]. Before proving Lemma 4.4, we require three preliminary lemmas.…”

“…G, B) is a transversal design with group size 13 and block size 4. Let S 1 = {(0, 1), (0, 2), (0, 3), (0, 4), (1, 1), (1,2), (1,3), (1,4), (2, 1), (2, 2), (2, 3), (2,4), (3,1), (3,2), (3,8), (3,9)}; S 2 = {(0, 5), (0, 6), (0, 7), (0, 8), (1,5), (1,6), (1,7), (1,10), (2,7), (2,8), (2,9), (2,12), (3, 0), (3,3), (3,10), (3,11)}; S 3 = {(0, 0), (0, 9), (0, 10), (0, 11), (1, 0), (1,8), (1,11), (1,12), (2, 0), (2,5), (2,6),…”

“…Then {S 1 , S 2 , S 3 } is a blocking system for (V, B), each set of which intersects each group in G in exactly 4 points. Let (1,6), (1,7), (1,9), (1,10), (2,8), (2,9), (2, 10), (2,11), (3, 0), (3,4), (3,5), (3,12)}; T 3 = {(0, 0), (0, 10), (0, 11), (0, 12), (1, 0), (1,8), (1,11), (1,12), (2, 0), (2,6), (2,7), (2,12), (3,6), (3,7), (3,8), (3,9)}.…”

“…Then {S 1 , S 2 , S 3 } is a blocking system for (V, B), each set of which intersects each group in G in exactly 4 points. Let T 1 = {(0, 2), (0, 3), (0, 4), (0, 5), (1, 2), (1,3), (1,4), (1,5), (2, 1), (2, 3), (2, 4), (2, 5), (3, 1), (3, 2), (3, 10), (3, 11)}; T 2 = {(0, 6), (0, 7), (0, 8), (0, 9), (1,6), (1,7), (1,9), (1,10), (2,8), (2,9), (2, 10), (2,11), (3, 0), (3,4), (3,5), (3, 12)}; T 3 = {(0, 0), (0, 10), (0, 11), (0, 12), (1, 0), (1,8), (1,11), (1,12), (2, 0), (2,6), (2,7), (2,12), (3,6), (3,7), (3,8), (3, 9)}.…”

Colourings of star systems

“…An m-cycle system of order n > 1 is a partition of the edges of the complete graph K n into m-cycles. Also, Horsley and Pike [5] showed that for all k 2 ⩾ and m 3 ⩾ with k m ( , ) (2,3) ≠ , there exist k-chromatic m-cycle systems of all admissible orders greater than or equal to some integer n k m , . A bipartite graph G m n , is a graph whose vertex set can be partitioned into two subsets V 1 and V 2 with m and n vertices each, such that every edge of G m n , has one vertex in V 1 and one vertex in V 2 .…”