Injective-edge-coloring of planar graphs with girth restriction.

A k - i n j e c t i v e - e d g e - c o l o r i n g o f a g r a p h G is a mapping f : E (G) → { 1 , 2 , ... , k } such that f (e 1) ≠ f (e 3) for any three consecutive edges e 1 , e 2 , e 3 of a path or a 3 -cycle. χ i ′ (G) = min { k | G has a k -injective-edge-coloring } is called the injective chromatic index of G. In this paper, we prove that for planar graphs G with Δ (G) ≥ 6 , (1) χ i ′ (G) ≤ 3 Δ (G) − 3 if g (G) ≥ 6 ; (2) χ i ′ (G) ≤ 3 Δ (G) − 4 if g (G) ≥ 7. [ABSTRACT FROM AUTHOR]