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A lower bound for on-line ranking number of a path. (English) Zbl 1115.05047
Summary: A $$k$$-ranking of a graph $$G$$ is a mapping $$\varphi:V(G)\to\{1, \dots,k\}$$ such that any path with endvertices $$x$$ and $$y$$ satisfying $$x\neq y$$ and $$\varphi(x)= \varphi(y)$$ contains a vertex $$z$$ with $$\varphi(z)>\varphi(x)$$. The ranking number $$\chi_r(G)$$ of $$G$$ is the minimum $$k$$ admitting a $$k$$-ranking of $$G$$. The on-line ranking number $$\chi^*_r(G)$$ of $$G$$ is the corresponding on-line invariant; in that case vertices of $$G$$ are coming one by one so that a partial ranking has to be chosen by considering only the structure of the subgraph of $$G$$ induced by the present vertices. It is known that $$\lfloor\log_2n\rfloor+1 =\chi_r(P_n)\leq\chi_r^*(P_n)\leq 2\lfloor\log_2n\rfloor+1$$. In this paper it is proved that $$\chi^*_r(P_n)>1.619\log_2n-1$$.

##### MSC:
 05C38 Paths and cycles
##### Keywords:
ranking; ranking number; on-line ranking number; path
Full Text:
##### References:
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