Abstract
AbstractThe concept of a (1, 2)‐eulerian weight was introduced and studied in several papers recently by Seymour, Alspach, Goddyn, and Zhang. In this paper, we proved that if G is a 2‐connected simple graph of order n (n ≧ 7) and w is a smallest (1, 2)‐eulerian weight of graph G, then |Ew=even | n ‐ 4, except for a family of graphs. Consequently, if G admits a nowhere‐zero 4‐flow and is of order at least 7, except for a family of graphs, the total length of a shortest cycle covering is at most | V(G) | + |E(G) |‐ 4. This result generalizes some previous results due to Bermond, Jackson, Jaeger, and Zhang.
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