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.

While recent computer vision algorithms achieve impressive performance on many benchmarks, they lack robustness - presented with an image from a different distribution, (e.g. weather or lighting conditions not considered during training), they may produce an erroneous prediction. Therefore, it is desired that such a model will be able to reliably predict its confidence measure. In this work, uncertainty estimation for the task of semantic segmentation is evaluated under a varying level of domain shift: in a cross-dataset setting and when adapting a model trained on data from the simulation. It was shown that simple color transformations already provide a strong baseline, comparable to using more sophisticated style-transfer data augmentation. Further, by constructing an ensemble consisting of models using different backbones and/or augmentation methods, it was possible to improve significantly model performance in terms of overall accuracy and uncertainty estimation under the domain shift setting. The Expected Calibration Error (ECE) on challenging GTA to Cityscapes adaptation was reduced from 4.05 to the competitive value of 1.1. Further, an ensemble of models was utilized in the self-training setting to improve the pseudo-labels generation, which resulted in a significant gain in the final model accuracy, compared to the standard fine-tuning (without ensemble).