Search

An edge coloring of a graph G is called interval edge coloring if for each v ��� V(G) the set of colors on edges incident to v forms an interval of integers. A graph G is interval colorable if there is an interval coloring of G. For an interval colorable graph G, by the interval chromatic index of G, denoted by ��'_i(G), we mean the smallest number k such that G is interval colorable with k colors. A bipartite graph G is called (��,��)-biregular if each vertex in one part has degree �� and each vertex in the other part has degree ��. A graph G is called (��*,��*)-bipartite if G is a subgraph of an (��,��)-biregular graph and the maximum degree in one part is �� and the maximum degree in the other part is ��. In the paper we study the problem of interval edge colorings of (k*,2*)-bipartite graphs, for k ��� {3,4,5}, and of (5*,3*)-bipartite graphs. We prove that every (5*,2*)-bipartite graph admits an interval edge coloring using at most 6 colors, which can be found in O(n^{3/2}) time, and we prove that an interval edge 5-coloring of a (5*,2*)-bipartite graph can be found in O(n^{3/2}) time, if it exists. We show that every (4^*,2^*)-bipartite graph admits an interval edge 4-coloring, which can be found in O(n) time. The two following problems of interval edge coloring are known to be NP-complete: 6-coloring of (6,3)-biregular graphs (Asratian and Casselgren (2006)) and 5-coloring of (5*,5*)-bipartite graphs (Giaro (1997)). In the paper we prove NP-completeness of 5-coloring of (5*,3*)-bipartite graphs., LIPIcs, Vol. 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021), pages 26:1-26:12

The hydrogen production technologies developed in the Institute of Fluid-Flow Machinery, Polish Academy of Sciences in Gdańsk are discussed here. They include the following methods: dark fermentation, photoelectrochemical water oxidation and hydrocarbons (or alcohols) reforming by microwave plasma. The potential of hydrogen production by using dark fermentation of different popular wastes such as: agricultural wastes, textile or wood waste, was determined using suitable models. Also, the influence of microaeration during dark fermentation of some substrates, e.g. sour cabbage, was tested. Photochemical oxidation is a water-splitting process driven by radiation at the surface of a titanium-oxide anode. The Si microrods covered by titania films were verified as a photoanode material. The hydrogen production from methane, ethanol, isopropanol and kerosene was driven by a microwave plasma. The results obtained confirm that microwave plasma sources have a high potential for hydrogen production via gaseous and liquid fuels reforming., Eco-Energetics: technologies, environment, law and economy, Tom 2 (2019)