Edge coloring of signed graphs

In this paper, we introduce edge coloring for signed graphs which is naturally corresponding to the vertex coloring of their signed line graphs. Let χ±′(G,σ) denote the edge chromatic number of a signed graph (G,σ). It follows from the definition that χ±′(G,σ)≥Δ, where Δ is the maximum degree of G. We attempt to establish Vizing type of theorem for χ±′(G,σ), and we are able to show that χ±′(G,σ)≤Δ+1 if Δ≤5 or if G is a planar graph. Further, we show that every planar graph with Δ=8 and without adjacent triangles has a linear 4-coloring, which confirm the Planar Linear Arboricity Conjecture for this family of graphs. A direct application of this result shows that χ±′(G,σ)=Δ if G is a planar graph with Δ≥10 or G is a planar graph with Δ∈{8,9} and without adjacent triangles.