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Every line graph of a 4-edge-connected graph is $$\mathbf Z_3$$-connected. (English) Zbl 1189.05086
Let $$G=(V,E)$$ be a digraph and let $$A$$ be a nontrivial additive abelian group with identity $$0$$. Let $$f$$ be a mapping from $$E$$ into $$A$$. Associated with $$f$$ is its boundary $$\partial f$$, a mapping from $$V$$ to $$A$$, defined by $$\partial f(x)=\sum_{e {\in E^{+}(x)}}f(e)-\sum_{e {\in E^{-}(x)}}f(e)$$. An undirected graph $$G$$ is $$A$$-connected if $$G$$ has an orientation such that for every $$b\colon V\to A$$ with $$\sum_{x\in V}b(x)=0$$ there is an $$f\colon E\to A \setminus\{0\}$$ with $$b=\partial f$$. A function $$f\colon E\to A \diagdown \{0\}$$ with $$\partial f = 0$$ is called an $$A$$-NZF (nowhere zero flow). A function $$f\colon E\to Z \setminus \{0\}$$, where $$Z$$ is the group of integers, with $$\partial f = 0$$ and such that for every $$e \in E$$, $$0 < |f(e)| < k$$ , is called a $$k$$-NZF. For a graph $$G$$ let $$\Lambda (G)$$ be the minimum $$k$$ such that if $$A$$ is an abelian group of order at least $$k$$, then $$G$$ is $$A$$-connected. W.T. Tutte [“A contribution to the theory of chromatic polynomials”, Canadian J. Math. 6, 80–91 (1954; Zbl 0055.17101)] conjectured that every 4-edge connected graph has a 3-NZF. Z.-H. Chen, H.-J. Lai, and H. Lai [“Nowhere zero flows in line graphs”, Discrete Math. 230, No. 1-3, 133–141 (2001; Zbl 0982.05080)] proved that if every 4-edge connected line graph has a 3-NZF, then every 4-edge connected graph has a 3-NZF. For a connected graph $$G$$, a partition $$(E_{1}, E_{2}, \dots , E_{n})$$ of the edges of $$G$$ is a $$(\geq 4)$$-clique partition provided $$G(E_{i})$$ is spanned by a complete graph with 4 or more vertices for $$i = 1,2, \dots, n$$.
The main result of this paper is if $$G$$ is 4-edge connected and $$G$$ has a $$(\geq 4)$$-clique partition, then $$\Lambda (G) \leq 3$$. As a consequence every line graph of a 4-edge connected graph has a 3-NZF.

##### MSC:
 05C40 Connectivity 05C35 Extremal problems in graph theory
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##### References:
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