Search

Let G be a graph with chromatic number χ(G) and consider a partition P of G into connected subgraphs. P is a puzzle on G if there is a unique vertex coloring of G using 1, 2, …, χ(G) such that the sums of the numbers assigned to the partition pieces are all the same. P is an apuzzle if there is a unique vertex coloring such that the sums are all different. We investigate the concept of puzzling and apuzzling graphs, detailing classes of graphs that are puzzling, apuzzling and neither.

In this investigation, 0.19–1.8 wt.% of Ge was introduced into a ternary Zn-Mg-Al alloy. The introduction of Ge had a significant impact on the microstructure, leading to the formation of Mg2Ge. The area fraction of the eutectic phase diminished with increasing Ge additions. Small-scale test techniques were utilised to evaluate the mechanical properties due to the changes in microstructure. Zn-Mg-Al alloys were found to be inherently harder compared to standard hot-dip Zn-containing 0.2 wt.% Al. The hardness and strength of the Zn-Mg-Al alloys decreased with the increase in Ge additions.