Their rancor dated from a political dogfight between them. A real dogfight for third place. A prolonged dogfight over their rival bids for the contract.
source: WordNet 3.0
disorderly fighting
source: WordNet 3.0
an aerial engagement between fighter planes
source: WordNet 3.0
a violent fight between dogs (sometimes organized illegally for entertainment and gambling)
source: WordNet 3.0
verb
arrange for an illegal dogfight
source: WordNet 3.0
engage in an aerial battle with another fighter plane