Vol. 8, No. 2, 2015

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On the $\varepsilon$-ascent chromatic index of complete graphs

Jean A. Breytenbach and C. M. (Kieka) Mynhardt

Vol. 8 (2015), No. 2, 295–305

An edge ordering of a graph G = (V,E) is an injection f : E +, where + is the set of positive integers. A path in G for which the edge ordering f increases along its edge sequence is called an f-ascent; an f-ascent is maximal if it is not contained in a longer f-ascent. The depression ε(G) of G is the smallest integer k such that any edge ordering f has a maximal f-ascent of length at most k. Applying the concept of ascents to edge colourings rather than edge orderings, we consider the problem of determining the minimum number χε(Kn) of colours required to edge colour Kn, n 4, such that the length of a shortest maximal ascent is equal to ε(Kn) = 3. We obtain new upper and lower bounds for χε(Kn), which enable us to determine χε(Kn) exactly for n = 7 and n 2(mod4) and to bound χε(K4m) by 4m χε(K4m) 4m + 1.

edge ordering of a graph, increasing path, depression, edge colouring
Mathematical Subject Classification 2010
Primary: 05C15, 05C78, 05C38
Received: 22 July 2013
Accepted: 26 October 2013
Published: 3 March 2015

Communicated by Jerrold Griggs
Jean A. Breytenbach
Computer Science, Department of Mathematical Sciences
Stellenbosch University
Private Bag X1
South Africa
C. M. (Kieka) Mynhardt
Department of Mathematics and Statistics
University of Victoria
P.O. Box 1700 STN CSC
Victoria, BC V8W 2Y2