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Double Eulerian Circuits in Multigraphs

By Valery Liskovets


We formulate explicitly and discuss a simple new enumerative formula for double (directed) eulerian circuits in n-edged labeled multigraphs. The formula follows easily from a recent 2-parametric formula of B. Lass. Multigraphs may have loops and are considered as symmetric multidigraphs. So, in an eulerian circuit, every edge is traversed exactly once in each direction (backtracks are allowed). With respect to undirected multigraphs such circuits are called here double eulerian circuits. A multigraph possesses a double eulerian circuit if and only if it is connected. We deal with labeled multigraphs, that is, multigraphs with numbered vertices. Moreover, any vertex may be distinguished as a root. If a multigraph is unrooted, then vertex 1 implicitly plays the role of the root. 1 Supported in part by the INTAS (Grant INTAS-BELARUS 97-0093) 1 All double eulerian circuits are considered as starting and finishing at the root. Two circuits are equivalent if they differ only in the order in which parallel edges (includin

Year: 2011
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