Sequential edge-coloring on the subset of vertices of almost regular graphs

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Let G be a graph and R V(G). A proper edge-coloring of a graph G with colors 1,,t is called an R-sequential t-coloring if the edges incident to each vertex v∈ R are colored by the colors 1,,dG(v), where dG(v) is the degree of the vertex v in G. In this note, we show that if G is a graph with Δ(G)-δ(G)≤ 1 and χ′(G)=Δ(G)=r (r≥ 3), then G has an R-sequential r-coloring with R ≥ (r-1)nᵣ+nr, where n= V(G) and nᵣ={v∈ V(G):dG(v)=r}. As a corollary, we obtain the following result: if G is a graph with Δ(G)-δ(G)≤ 1 and χ′(G)=Δ(G)=r (r≥ 3), then Σ′(G)≤ 2nᵣ(2r-1)+n(r-1)(r²+2r-2)4r, where Σ′(G) is the edge-chromatic sum of G.