Recently, Behr (2020) introduced a notion of the chromatic index of signed graphs and proved that for every signed graph ( G , σ ) it holds that Δ ( G ) ≤ χ ′ ( G , σ ) ≤ Δ ( G ) + 1 , where Δ ( G ) is the maximum degree of G and χ ′ denotes its chromatic index. In general, the chromatic index of ( G , σ ) depends on both the underlying graph G and the signature σ . In the paper we study graphs G for which χ ′ ( G , σ ) does not depend on σ . To this aim we introduce two new classes of graphs, namely 1 ± and 2 ± , such that graph G is of class 1 ± (respectively, 2 ± ) if and only if χ ′ ( G , σ ) = Δ ( G ) (respectively, χ ′ ( G , σ ) = Δ ( G ) + 1 ) for all possible signatures σ . We prove that all wheels, necklaces, complete bipartite graphs K r , t with r ≠ t and almost all cacti graphs are of class 1 ± . Moreover, we give sufficient and necessary conditions for a graph to be of class 2 ± , i.e. we show that these graphs must have odd maximum degree and give examples of such graphs with arbitrary odd maximum degree bigger than 1.

In the paper we introduce a new variant of the graph coloring game and a new graph parameter being the result of the new game. We study their properties and get some lower and upper bounds, exact values for complete multipartite graphs and optimal, often polynomial-time strategies for both players provided that the game is played on a graph with an odd number of vertices. At the end we show that both games, the new and the classic one, are related: our new parameter is an upper bound for the game chromatic number.