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Let $G=(V(G), E(G))$ be a multigraph with maximum degree $\Delta(G)$, chromatic index $\chi'(G)$ and total chromatic number $\chi''(G)$. The Total Coloring conjecture proposed by Behzad and Vizing, independently, states that $\chi''(G)\leq \Delta(G)+\mu(G) +1$ for a multigraph $G$, where $\mu(G)$ is the multiplicity of $G$. Moreover, Goldberg conjectured that $\chi''(G)=\chi'(G)$ if $\chi'(G)\geq \Delta(G)+3$ and noticed the conjecture holds when $G$ is an edge-chromatic critical graph. By assuming the Goldberg-Seymour conjecture, we show that $\chi''(G)=\chi'(G)$ if $\chi'(G)\geq \max\{ \Delta(G)+2, |V(G)|+1\}$ in this note. Consequently, $\chi''(G) = \chi'(G)$ if $\chi'(G) \ge \Delta(G) +2$ and $G$ has a spanning edge-chromatic critical subgraph.

Let $G$ be a simple graph, and let $n$, $\Delta(G)$ and $\chi' (G)$ be the order, the maximum degree and the chromatic index of $G$, respectively. We call $G$ overfull if $|E(G)|/\lfloor n/2\rfloor > \Delta(G)$, and critical if $\chi'(H) < \chi'(G)$ for every proper subgraph $H$ of $G$. Clearly, if $G$ is overfull then $\chi'(G) = \Delta(G)+1$. The core of $G$, denoted by $G_{\Delta}$, is the subgraph of $G$ induced by all its maximum degree vertices. Hilton and Zhao conjectured that for any critical class 2 graph $G$ with $\Delta(G) \ge 4$, if the maximum degree of $G_{\Delta}$ is at most two, then $G$ is overfull, which in turn gives $\Delta(G) > n/2 +1$. We show that for any critical class 2 graph $G$, if the minimum degree of $G_{\Delta}$ is at most two and $\Delta(G) > n/2 +1$, then $G$ is overfull.

For a given graph H , the Ramsey number r ( H ) is the minimum N such that any 2-edge-coloring of the complete graph K N yields a monochromatic copy of H . Given a positive integer n , a fan F n is a graph formed by n triangles that share one common vertex. We show that 9 n ∕ 2 − 5 ≤ r ( F n ) ≤ 11 n ∕ 2 + 6 for any n . This improves previous best bounds r ( F n ) ≤ 6 n of Lin and Li and r ( F n ) ≥ 4 n + 2 of Zhang, Broersma and Chen.

In this study, it was found that an ancient bronze sword had special microstructures, i.e., a tin (Sn)-rich layer (Sn: 38.51 wt.%), that was around 0.1–0.3 mm in thickness in the bronze substrate (Sn: 18.57 wt.%). This sword was unearthed from the same Chu tombs of the “Sword of Gou Jian”, and dated back to the late Spring and Autumn Period (496 BC–464 BC). The experimental and theoretical analyses revealed that (1) the Sn-rich layer exhibited higher microhardness (around 650 HV) than the sword body (around 300 HV); (2) the Sn-rich layer showed a brittle fracture due to the formation of a large amount of α + δ eutectoid, while the sword body was of good toughness due to a large amount of α-Cu solid solution phase; and (3) theoretical calculations of Sn diffusion in the Cu substrate indicated that this Sn-rich layer could have been formed within several hours or several days if the temperature was above 600 °C. Therefore, this sword was proposed to be a novel kind of composite bronze sword, and the possible manufacturing technique was a surface treatment called “dip or wipe tinning” or tin amalgam, which was widely used in the Bronze Age. Technically, this process possesses more advantages than the well-known two-times casting for making the “double-colour” or bi-metallic composite bronze sword. This research showed that the materials processing level was beyond our expectations for ancient China 2500 years ago.

The structures and energetics of the cyclopentadienyliron carbonyl boronyls Cp2Fe2(BO)2(CO)n (n = 3, 2, 1, 0) have been examined by density functional theory. The lowest energy Cp2Fe2(BO)2(CO)n (n = 2, 0) structures are found to have bridging dioxodiborene (µ-B2O2) ligands that can exist as cis or trans stereoisomers. Either µ-B2O2 stereoisomer functions as a net three-electron donor to each iron atom in such binuclear complexes. The lowest energy Cp2Fe2(BO)2(CO)n (n = 3, 1) structures resemble their isoelectronic Cp2Mn2(CO)n+2 analogues. Thus the lowest energy Cp2Fe2(BO)2(CO)3 structure consists of two CpFe(BO)(CO) units joined by an Fe–Fe bond bridged by a CO group. The lowest energy Cp2Fe2(BO)2(CO) structure is a singlet triply bridged structure with a short ∼2.17 A Fe Fe distance suggesting a formal triple bond. This latter structure is very similar to the experimental structure of the isoelectronic Cp*2Mn2(µ-CO)3, synthesized by Hermann and coworkers and shown by X-ray crystallography to be a triply bridged structure with a ∼2.17 A Mn Mn distance.