## Abstract

Let G be a graph, let Γ be an Abelian group with identity 0Γ, and, for each vertex v of G, let p(v) be a prescription such that ∑

∑e

If such an orientation D and labelling f exists for all such p,then G is Γ -connected.

Our main result is that if G

is a 5-edge-connected planar graph and |Γ|≥3, then G is Γ-connected. This is equivalent to a dual colourability statement proved by Lai and Li (2007): planar graphs with girth at least 5 are “Γ-colourable”. Our proof is considerably shorter than theirs. Moreover, the Γ -colourability result of Lai and Li is already a consequence of Thomassen’s (2003) 3-list-colour proof for planar graphs of girth at least 5.

Our theorem (as well as the girth 5 colourability result) easily implies that every 5-edge-connected planar graph for which |E(G)|

is a multiple of 3 has a claw decomposition, resolving a question of Barát and Thomassen. It also easily implies the dual of Grötzsch’s Theorem, that every planar graph without 1- or 3-cut has a 3-flow; this is equivalent to Grötzsch’s Theorem.

_{v∈V(G)}p(v)=0Γ. A (Γ,p)-flow consists of an orientation D of G and, for each edge e of G, a label f(e) in Γ∖{0Γ} such that, for each vertex v of G,∑e

_{points in to v}f(e)−∑e_{points out from v}f(e)=p(v)If such an orientation D and labelling f exists for all such p,then G is Γ -connected.

Our main result is that if G

is a 5-edge-connected planar graph and |Γ|≥3, then G is Γ-connected. This is equivalent to a dual colourability statement proved by Lai and Li (2007): planar graphs with girth at least 5 are “Γ-colourable”. Our proof is considerably shorter than theirs. Moreover, the Γ -colourability result of Lai and Li is already a consequence of Thomassen’s (2003) 3-list-colour proof for planar graphs of girth at least 5.

Our theorem (as well as the girth 5 colourability result) easily implies that every 5-edge-connected planar graph for which |E(G)|

is a multiple of 3 has a claw decomposition, resolving a question of Barát and Thomassen. It also easily implies the dual of Grötzsch’s Theorem, that every planar graph without 1- or 3-cut has a 3-flow; this is equivalent to Grötzsch’s Theorem.

Original language | English |
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Journal | The Electronic Journal of Combinatorics |

Volume | 7 |

Issue number | 2-3 |

Pages (from-to) | 219-232 |

ISSN | 1097-1440 |

DOIs | |

Publication status | Published - 2016 |