Avoidance colourings for small nonclassical Ramsey numbers
Date
2011
Authors
Burger A.P.
Van Vuuren J.H.
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Abstract
The irredundant Ramsey number s = s(m; n) [upper domination Ramsey number u = u(m; n), respectively] is the smallest natural number s [u, respectively] such that in any red-blue edge colouring (R;B) of the complete graph of order s [u, respectively], it holds that IR(B) ≥m or IR(R) ≥n [ (B) ≥m or (R) ≥n, respectively], where and IR denote respectively the upper domination number and the irredundance number of a graph. Furthermore, the mixed irredundant Ramsey number t = t(m; n) [mixed domination Ramsey number v = v(m; n), respectively] is the smallest natural number t [v, respectively] such that in any red-blue edge colouring (R;B) of the complete graph of order t [v, respectively], it holds that IR(B) ≥m or β(R) ≥n [τ(B) ≥m or β (R) ≥n, respectively], where β denotes the independent domination number of a graph. These four classes of non-classical Ramsey numbers have previously been studied in the literature. In this paper we introduce a new Ramsey number w = w(m; n), called the irredundant-domination Ramsey number, which is the smallest natural number w such that in any red-blue edge colouring (R;B) of the complete graph of order w, it holds that IR(B) ≥m or (R) ≥n. A computer search is employed to determine complete sets of avoidance colourings of small order for these five classes of nonclassical Ramsey numbers. In the process the fifteen previously unknown Ramsey numbers t(4; 4) = 14, t(6; 3) = 17, u(4; 4) = 13, v(4; 3) = 8, v(4; 4) = 14, v(5; 3) = 13, v(6; 3) = 17, w(3; 3) = 6, w(3; 4) = w(4; 3) = 8, w(4; 4) = 13, w(3; 5) = w(5; 3) = 12 and w(3; 6) = w(6; 3) = 15 are established. © 2011 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.
Description
Keywords
Complete graphs, Complete sets, Computer search, Edge-colouring, Extremal colouring., Independent domination number, Irredundance number, Mixed Ramsey, Natural number, Nonclassical Ramsey number, Ramsey numbers, Upper domination numbers, Graph theory
Citation
Discrete Mathematics and Theoretical Computer Science
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