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Let $N\geq 2$ be an integer, a (1, $N$)-periodic graph $G$ is a periodic graph whose vertices can be partitioned into two sets $V_1=\{v\mid\sigma(v)=v\}$ and $V_2=\{v\mid\sigma^i(v)\neq v\ \mbox{for any}\ 1

Given a graph $G$, denote by $\Delta$, $\bar{d}$ and $\chi^\prime$ the maximum degree, the average degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$\Delta$-critical} if $\chi^\prime(G)=\Delta+1$ and $\chi^\prime(H)\le\Delta$ for every proper subgraph $H$ of $G$. Vizing in 1968 conjectured that if $G$ is edge-$\Delta$-critical, then $\bar{d}\geq \Delta-1+ \frac{3}{n}$. We show that $$ \begin{displaystyle} \avd \ge \begin{cases} 0.69241\D-0.15658 \quad\,\: \mbox{ if } \Delta\geq 66, 0.69392\D-0.20642\quad\;\,\mbox{ if } \Delta=65, \mbox{ and } 0.68706\D+0.19815\quad\! \quad\mbox{if } 56\leq \Delta\leq64. \end{cases} \end{displaystyle} $$ This result improves the best known bound $\frac{2}{3}(\Delta +2)$ obtained by Woodall in 2007 for $\Delta \geq 56$. Additionally, Woodall constructed an infinite family of graphs showing his result cannot be improved by well-known Vizing's Adjacency Lemma and other known edge-coloring techniques. To over come the barrier, we follow the recently developed recoloring technique of Tashkinov trees to expand Vizing fans technique to a larger class of trees., Comment: 23 pages

Offshore oil and gas exploration has become the main field at present and in the future; however, the information about offshore deepwater exploration is less than onshore. At present, the offshore oil and gas exploration is mainly based on seismic data, thus, it is significant to use seismic data to detect hydrocarbon in offshore deepwater exploration. Using seismic data to do hydrocarbon detection is a process of inversion, and it has ambiguity and uncertainty. This paper illustrated five methods to detect hydrocarbon, namely, seismic amplitude attribute, frequency attribute, spectrum decomposition method, waveform classification and cross plot analysis of far offset stack data and near offset stack data. The combined application of those methods can greatly reduce the ambiguity and uncertainty. According to the study of deepwater sediment in Equatorial Guinea, the results of hydrocarbon detection coincide with drilled wells and the known hydrocarbon distribution. The combined using of those methods has achieved better effect in this study area, and has formed a series of methods of hydrocarbon detection in offshore deepwater oil and gas exploration.