Circuit Decompositions of Eulerian Graphs

Abstract

Let G be an eulerian graph. For each vertex v ∈ V ( G ), let P ( v ) be a partition of the edges incident with v and set P =∪ v ∈ V ( G ) P ( v ), called a forbidden system of G . We say that P is admissible if | P ∩ T |⩽ 1 2 | T | for every P ∈ P and every edge cut T of G . H. Fleischner and A. Frank (1990, J. Combin. Theory Ser. B 50 , 245–253) proved that if G is planar and P is any admissible forbidden system of G , then G has a circuit decomposition F such that | C ∩ P |⩽1 for every C ∈ F and every P ∈ P . We generalize this result to all eulerian graphs that do not contain K 5 as a minor. As a consequence, a conjecture of Sabidussi is settled for graphs that do not contain K 5 as a minor. Also, as a byproduct, our proof provides a different approach to the circuit cover theorem of B. Alspach, L. A. Goddyn, and C.-Q. Zhang (1994, Trans. Amer. Math. Soc. 344 , No. 1, 131–154).