A coloring of edges a graph G is injective if for any two distinct e1 and e2, the colors e2 are they at distance 1 in or common triangle. Naturally, chromatic index G, ?inj?(G), minimum number needed an edge-coloring G. We study how large can be terms maximum degree when we have restrictions on girth and/or also compare our bounds with analogous strong index.

Let G be a (simple) graph with maximum degree three and chromatic index four. A 3-edge-coloring of G is a coloring of its edges in which only three colors are used. Then a vertex is conflicting when some edges incident to it have the same color. The minimum possible number of conflicting vertices that a 3edge-coloring of G can have, d(G), is called the edge-coloring degree of G. Here we are mai...

We conjecture that any graphG with treewidth k and maximum degree ∆(G) ≥ k + √ k satisfies χ′(G) = ∆(G). In support of the conjecture we prove its fractional version.

For a graph G, denote by fk(G) the smallest number of vertices that must be deleted from G so that the remaining induced subgraph has its maximum degree shared by at least k vertices. It is not difficult to prove that there are graphs for which already f2(G) ≥ √ n(1− o(1)), where n is the number of vertices of G. It is conjectured that fk(G) = Θ( √ n) for every fixed k. We prove this for k = 2,...