A three-dimensional mathematical and physical model coupling with the heat transfer and the flow of molten metal in the centrifugal casting of the high speed steel roll was established by using CFD software FLUENT. It can be used to analyze the distribution of the temperature filed and the flow filed in the centrifugal casting under the gravity, the electromagnetic stirring force and the centrifugal force. Some experiments were carried out to verify the above analysis results. The effects of the electromagnetic force on the centrifugal casting process are discussed. The results showed that under the 0.15 T electromagnetic field intensity, both the absolute pressure of metal flow to mold wall and the metal flow velocity on the same location have some differences between the electromagnetic centrifugal casting and the centrifugal casting. Numerical results for understanding the electromagnetic stirring of the centrifugal casting process have a guiding significance.

Abstract:
The as-cast microstructures of high carbon high speed steels (HC-HSS) made by sand casting, centrifugal casting and electromagnetic centrifugal casting, respectively, were studied by using of optical microscopy (OM) and D/max2200pc X-ray diffraction. The results show that the microstructure of as-cast HC-HSS is dominated by alloy carbides (W2C, VC, Cr7C3), martensite and austenite. The centrifugal casting and electromagnetic centrifugal casting apparently improve the solidification structure of HC-HSS. With the increase of magnetic intensity (B), the volume fraction of austenite in the HC-HSS solidification structure increases significantly while the eutectic ledeburite decreases. Moreover, the secondary carbides precipitated from the austenite are finer with more homogeneous distribution in the electromagnetic centrifugal castings. It has also been found that the lath of eutectic carbide in ledeburite becomes finer and carbide phase spacing in eutectic ledeburite increases along with the higher magnetic field strength.

Abstract:
This paper proves that for any positive integer $k$, every essentially $(2k+1)$-unbalanced $(12k-1)$-edge connected signed graph has circular flow number at most $2+\frac 1k$.

Abstract:
This paper introduces a new variant of hypercubes, which we call Z-cubes. The n-dimensional Z-cube $H_n$ is obtained from two copies of the (n-1)-dimensional Z-cube $H_{n-1}$ by adding a special perfect matching between the vertices of these two copies of $H_{n-1}$. We prove that the n-dimensional Z-cubes $H_n$ has diameter $(1+o(1))n/\log_2 n$. This greatly improves on the previous known variants of hypercube of dimension n, whose diameters are all larger than n/3. Moreover, any hypercube variant of dimension $n$ is an n-regular graph on $2^n$ vertices, and hence has diameter greater than $n/\log_2 n$. So the Z-cubes are optimal with respect to diameters, up to an error of order $o(n/\log_2n)$. Another type of Z-cubes $Z_{n,k}$ which have similar structure and properties as $H_n$ are also discussed in the last section.

Abstract:
Given any rational numbers $r geq r' >2$ and an integer $g$, we prove that there is a graph $G$ of girth at least $g$, which is uniquely $r$-colourable and uniquely $r'$-fractional colourable.

Abstract:
This paper studies the choice number and paint number of the lexicographic product of graphs. We prove that if $G$ has maximum degree $\Delta$, then for any graph $H$ on $n$ vertices $ch(G[H]) \le (4\Delta+2)(ch(H) +\log_2 n)$ and $\chi_P(G[H]) \le (4\Delta+2) (\chi_P(H)+ \log_2 n)$.

Abstract:
A graph $G$ is $(k,k')$-choosable if the following holds: For any list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and assigns to each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a total weighting $\phi: V(G) \cup E(G) \to R$ such that $\phi(z) \in L(z)$ for $z \in V \cup E$, and $\sum_{e \in E(u)}\phi(e)+\phi(u) \ne \sum_{e \in E(v)}\phi(e)+\phi(v)$ for every edge $uv$. This paper proves the following results: (1) If $G$ is a connected $d$-degenerate graph, and $k>d$ is a prime number, and $G$ is either non-bipartite or has two non-adjacent vertices $u,v$ with $d(u)+d(v) < k$, then $G$ is $(1,k)$-choosable. As a consequence, every planar graph with no isolated edges is $(1,7)$-choosable, and every connected $2$-degenerate non-bipartite graph other than $K_2$ is $(1,3)$-choosable. (2) If $d+1$ is a prime number, $v_1, v_2, \ldots, v_n$ is an ordering of the vertices of $G$ such that each vertex $v_i$ has back degree $d^-(v_i) \le d$, then there is a graph $G'$ obtained from $G$ by adding at most $d-d^-(v_i)$ leaf neighbours to $v_i$ (for each $i$) and $G'$ is $(1,2)$-choosable. (3) If $G$ is $d$-degenerate and $d+1$ a prime, then $G$ is $(d,2)$-choosable. In particular, $2$-degenerate graphs are $(2,2)$-choosable. (4) Every graph is $(\lceil\frac{{\rm mad}(G)}{2}\rceil+1, 2)$ -choosable. In particular, planar graphs are $(4,2)$-choosable, planar bipartite graphs are $(3,2)$-choosable.

Abstract:
A total weighting of a graph $G$ is a mapping $f$ which assigns to each element $z \in V(G) \cup E(G)$ a real number $f(z)$ as its weight. The vertex sum of $v$ with respect to $f$ is $\phi_f(v)=\sum_{e \in E(v)}f(e)+f(v)$. A total weighting is proper if $\phi_f(u) \ne \phi_f(v)$ for any edge $uv$ of $G$. A $(k,k')$-list assignment is a mapping $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ permissible weights, and assigns to each edge $e$ a set $L(e)$ of $k'$ permissible weights. We say $G$ is $(k,k')$-choosable if for any $(k,k')$-list assignment $L$, there is a proper total weighting $f$ of $G$ with $f(z) \in L(z)$ for each $z \in V(G) \cup E(G)$. It was conjectured in [T. Wong and X. Zhu, Total weight choosability of graphs, J. Graph Theory 66 (2011), 198-212] that every graph is $(2,2)$-choosable and every graph with no isolated edge is $(1,3)$-choosable. A promising tool in the study of these conjectures is Combinatorial Nullstellensatz. This approach leads to conjectures on the permanent indices of matrices $A_G$ and $B_G$ associated to a graph $G$. In this paper, we establish a method that reduces the study of permanent of matrices associated to a graph $G$ to the study of permanent of matrices associated to induced subgraphs of $G$. Using this reduction method, we show that if $G$ is a subcubic graph, or a $2$-tree, or a Halin graph, or a grid, then $A_G$ has permanent index $1$. As a consequence, these graphs are $(2,2)$-choosable. \end{abstract} {\small \noindent{{\bf Key words: } Permanent index, matrix, total weighting}

Abstract:
A graph G is called $(a:b)$-choosable if for any list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $a$ permissible colors, there is a $b$-tuple $L$-coloring of $G$. An $(a:1)$-choosable graph is also called $a$-choosable. In the pioneering paper on list coloring of graphs by Erd\H{o}s, Rubin, and Taylor, 2-choosable graphs are characterized. Confirming a special case of a conjecture in that paper, Tuza and Voigt proved that 2-choosable graphs are $(2m:m)$-choosable for any positive integer $m$. On the other hand, Voigt proved that if $m$ is an odd integer, then these are the only $(2m:m)$-choosable graphs; however, when $m$ is even, there are $(2m:m)$-choosable graphs that are not 2-choosable. A graph is called 3-choosable-critical if it is not 2-choosable, but all its proper subgraphs are $2$-choosable. Voigt conjectured that for every positive integer $m$, all 3-choosable-critical bipartite graphs are $(4m:2m)$-choosable. In this paper, we determine which 3-choosable-critical graphs are $(4:2)$-choosable, refuting Voigt's conjecture in the process. Nevertheless, a weaker version of the conjecture is true: we prove that there is an even integer $k$ such that for any positive integer $m$, every 3-choosable-critical bipartite graph is $(2km:km)$-choosable. Moving beyond 3-choosable-critical graphs, we present an infinite family of non-3-choosable-critical graphs which have been shown by computer analysis to be $(4:2)$-choosable, which shows that the family of all $(4:2)$-choosable graphs has rich structure.

Abstract:
The on-line choice number of a graph is a variation of the choice number defined through a two person game. It is at least as large as the choice number for all graphs and is strictly larger for some graphs. In particular, there are graphs $G$ with $|V(G)| = 2 \chi(G)+1$ whose on-line choice numbers are larger than their chromatic numbers, in contrast to a recently confirmed conjecture of Ohba that every graph $G$ with $|V(G)| \le 2 \chi(G)+1$ has its choice number equal its chromatic number. Nevertheless, an on-line version of Ohba conjecture was proposed in [P. Huang, T. Wong and X. Zhu, Application of polynomial method to on-line colouring of graphs, European J. Combin., 2011]: Every graph $G$ with $|V(G)| \le 2 \chi(G)$ has its on-line choice number equal its chromatic number. This paper confirms the on-line version of Ohba conjecture for graphs $G$ with independence number at most 3. We also study list colouring of complete multipartite graphs $K_{3\star k}$ with all parts of size 3. We prove that the on-line choice number of $K_{3 \star k}$ is at most $3/2k$, and present an alternate proof of Kierstead's result that its choice number is $\lceil (4k-1)/3 \rceil$. For general graphs $G$, we prove that if $|V(G)| \le \chi(G)+\sqrt{\chi(G)}$ then its on-line choice number equals chromatic number.