(d, 1)-total labeling of lexicographic products of some classes of graphs
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摘要: 主要讨论路Pn和Pm、路Pn和圈Cn的字典式乘积图的（d，1）-全标号，得出字典式乘积图${P_n} \circ {P_m}$、${P_n} \circ {C_m}$在一定约束条件下的（d，1）-全数λdT（G）的确切值.
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关键词:
- 字典式乘积 /
- (d, 1)-全标号 /
- (d, 1)-全数λdT(G)
Abstract: This paper focuses on (d, 1)-total labeling of the lexicographic products of path Pn and path Pm, path Pn and circle Cn, and gets the exact value of the (d, 1)-total number λdT (G) of lexicographic product ${P_n} \circ {P_m}$, ${P_n} \circ {C_m}$ under certain constraints. -
表 1 当$n$为偶数时, $ {P_2\circ P_n }$的一种${(n+4)}- {(2, 1)}$-全标号方法
Tab. 1 $(n+4)-(2, 1)$-total labelling of $P_2 \circ P_n $, when $n$ is even
$x_{11} (0)$ $x_{12} (1)$ $\cdots $ $x_{1(n-1)} (0)$ $x_{1n} (1)$ $x_{21} (n+3)$ 2 3 $\cdots $ $n$ $n+1$ $x_{22} (n+4)$ 3 4 $\cdots $ $n+1$ $n+2$ $\vdots $ $\vdots $ $\vdots $ $\vdots $ $\vdots $ $x_{2(n-1)} (n+3)$ $n$ $n+1$ $\cdots $ $n-2$ $n-1$ $x_{2n} (n+4)$ $n+1$ $n+2$ $\cdots $ $n-1$ $n$ 
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表 2 当$n$为奇数时, ${P_2\circ P_n }$的一种$ {(n+4)-(2, 1)}$-全标号方法
Tab. 2 $(n+4)-(2, 1)$-total labelling of $P_2 \circ P_n $, when $n$ is even odd
$x_{11} (0)$ $x_{12} (1)$ $\cdots $ $x_{1(n-1)} (1)$ $x_{1n} (0)$ $x_{21} (n+4)$ 2 3 $\cdots $ $n$ $n+1$ $x_{22} (n+3)$ 3 4 $\cdots $ $n+1$ 2 $\vdots $ $\vdots $ $\vdots $ $\vdots $ $\vdots $ $x_{2(n-1)} (n+3)$ $n$ $n+1$ $\cdots $ $n-2$ $n-1$ $x_{2n} (n+4)$ $n+1$ $n+2$ $\cdots $ $n-1$ $n$
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表 3 当$n$为偶数时${P_2 \circ P_n} $的一种${(\Delta +d+2)-(d, 1)}$-全标号方法
Tab. 3 $(\Delta+d+2)-(d, 1)$-total labelling of $P_2 \circ P_n $, when $n$ is even
$x_{11} (0)$ $x_{12} (1)$ $x_{13} (0)$ $\cdots $ $x_{1(n-1)}(0)$ $x_{1n} (1)$ $x_{21} (3)$ $d+4$ $d+5$ $d+6$ $\cdots $ $d+\Delta $ $d+\Delta +1$ $x_{22} (2)$ $d+5$ $d+6$ $d+7$ $\cdots $ $d+\Delta +1$ $d+\Delta +2$ $x_{23} (3)$ $d+6$ $d+7$ $d+8$ $\cdots $ $d+\Delta +2$ $d+5$ $\vdots $ $\vdots $ $\vdots $ $\vdots $ $\vdots $ $\vdots $ $x_{2(n-1)} (3)$ $d+\Delta $ $d+\Delta +1$ $d+\Delta +2$ $\cdots $ $d+\Delta -2$ $d+\Delta -1$ $x_{2n} (2)$ $d+\Delta +1$ $d+\Delta +2$ $d+5$ $\cdots $ $d+\Delta -1$ $d+\Delta $
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表 4 ${P_2 \circ P_3}$的一种${(\Delta +d+1)-(d, 1)}$-全标号方法
Tab. 4 $(\Delta +d+1)-(d, 1)$-total labelling of $P_2 \circ P_3$
$x_{11} (2)$ $x_{12} (0)$ $x_{13} (2)$ $x_{21} (3)$ $d+4$ $d+5$ $d+6$ $x_{22} (1)$ $d+2$ $d+1$ $d+5$ $x_{23} (3)$ $d+5$ $d+6$ $d+3$
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表 5 $n$为偶数时, ${P_2 \circ C_n}$的一种${(\Delta +d+2)-(d, 1)}$-全标号方法
Tab. 5 $(\Delta+d+2)-(d, 1)$-total labelling of $P_2 \circ C_n $, when $n$ is even
$x_{11} (0)$ $x_{12} (1)$ $x_{13} (0)$ $\cdots $ $x_{1(n-1)}(0)$ $x_{1n} (1)$ $x_{21} (3)$ $d+2$ $d+5$ $d+6$ $\cdots $ $d+\Delta $ $d+\Delta +1$ $x_{22} (2)$ $d+5$ $d+6$ $d+7$ $\cdots $ $d+\Delta +1$ $d+\Delta +2$ $x_{23} (3)$ $d+6$ $d+7$ $d+8$ $\cdots $ $d+\Delta +2$ $d+5$ $\vdots $ $\vdots $ $\vdots $ $\vdots $ $\vdots $ $\vdots $ $x_{2(n-1)} (3)$ $d+\Delta $ $d+\Delta +1$ $d+\Delta +2$ $\cdots $ $d+\Delta -2$ $d+\Delta -1$ $x_{2n} (2)$ $d+\Delta +1$ $d+\Delta +2$ $d+5$ $\cdots $ $d+\Delta -1$ $d+\Delta $
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表 6 $n$为奇数时, ${P_2 \circ C_n }$的一种${(\Delta +d+{4})-(d, 1)}$-全标号方法
Tab. 6 $(\Delta+d+2)-(d, 1)$-total labelling of $P_2 \circ C_n $, when $n$ is odd
$x_{11} (0)$ $x_{12} (1)$ $x_{13} (0)$ $\cdots $ $x_{1(n-1)}(1)$ $x_{1n} (4)$ $x_{21} (2)$ $d+5$ $d+6$ $d+7$ $\cdots $ $d+\Delta +1$ $d+\Delta +2$ $x_{22} (3)$ $d+6$ $d+7$ $d+8$ $\cdots $ $d+\Delta +2$ $d+\Delta +3$ $x_{23} (2)$ $d+7$ $d+8$ $d+9$ $\cdots $ $d+\Delta +3$ $d+\Delta +4$ $\vdots $ $\vdots $ $\vdots $ $\vdots $ $\vdots $ $\vdots $ $x_{2(n-1)} (3)$ $d+\Delta +1$ $d+\Delta +2$ $d+\Delta+3$ $\cdots $ $d+\Delta -1$ $d+\Delta $ $x_{2n} (5)$ $d+\Delta +2$ $d+\Delta +3$ $d+6$ $\cdots $ $d+\Delta $ $d+\Delta +1$
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