JOURNAL OF COMBINATORIAL THEORY, Series B 50, X5-253 (1990) On Circuit Decomposition of Planar Eulerian Graphs* HERBERT FLEKCHNER Institur fir Informationsverarbeirung, Domus Universitetis, Wien, i5sterreiehische. Akademie der Wissenschaften AND ANDRAS FRANK Department of Computer Science, E&viis Lo&d University, Budapest Communicated by the Managing Editors July 27, 1988 We give a common generalization of P. Seymour’s “Integer sum of circuits” theorem and the tirst author’s theorem on decomposition of planar Eulerian graphs into circuits without forbidden transitions. “,I 1990 Academic Press. Inc. 1. INTRODUCTION It is well known that a non-negative integer-valued circulation can always be expressed as a non-negative integer combination of (incidence vectors of) directed circuits. Thus Hoffman’s circulation theorem (See, e.g., [2]) can be interpreted as one giving a necessary and sufficient condition for the existence of a list of directed circuits of a digraph so that the number of circuits from the list containing any edge is between two integer bounds given in advance. P. Seymour [3] proved the undirected counterpart of Hoffman’s result. THEOREM 1.1. Let G = (V, E) be an undirected graph endowed with two functions f, g: E + R + for which f < g. There are non-negative variables x(C) assigned to the circuits C of G for which f (e) < .X(x(C) : C a circuit and * Work supported by Sonderforschungsbereich 303 (DFG). Institut fiir Operations Research. Universitit Bonn. 245 582W50!2-8 0095~8956190 $3.00 Copyright C: 1990 by Academic Press, Inc. All rights ol reproducfion III any form reserved.