奇循环数的猜测
Guessing Numbers of Odd Cycles
摘要：
对于给定的颜色数$s$，图的猜测数是图的顶点集的最大着色族的大小的底$s$对数，使得每个顶点的颜色可以从其邻域中的顶点的颜色确定。$n$ -顶点圈图$C_n$的猜测数的上界是$n/2$ .已知只要$n$是偶数或$s$是完全正方形\cite{Christofides2011guessing}，猜测数就等于$n/2$。我们证明了，对于任意给定的整数$s\geq 2$，如果$a$是$s$小于等于$\sqrt{s}$的最大因子，对于足够大的奇数$n$，$C_n$与$s$颜色的猜测数是$(n-1)/2 + \log_s(a)$ .这回答了Christofides和Markström在2011年\cite{Christofides2011guessing}中提出的一个问题。我们还提出了一个明确的协议，实现这一界限的每一个$n$。将其与具有边信息的索引编码相联系，我们推断出具有$s$颜色的$C_n$的信息缺陷对于足够大的奇数$n$是$(n+1)/2 - \log_s(a)$。我们的研究结果是一个概括的$s=2$的情况下，这是证明在\cite{bar2011index}。
英文摘要：
For a given number of colours, $s$, the guessing number of a graph is the base $s$ logarithm of the size of the largest family of colourings of the vertex set of the graph such that the colour of each vertex can be determined from the colours of the vertices in its neighbourhood. An upper bound for the guessing number of the $n$-vertex cycle graph $C_n$ is $n/2$. It is known that the guessing number equals $n/2$ whenever $n$ is even or $s$ is a perfect square \cite{Christofides2011guessing}. We show that, for any given integer $s\geq 2$, if $a$ is the largest factor of $s$ less than or equal to $\sqrt{s}$, for sufficiently large odd $n$, the guessing number of $C_n$ with $s$ colours is $(n-1)/2 + \log_s(a)$. This answers a question posed by Christofides and Markström in 2011 \cite{Christofides2011guessing}. We also present an explicit protocol which achieves this bound for every $n$. Linking this to index coding with side information, we deduce that the information defect of $C_n$ with $s$ colours is $(n+1)/2 - \log_s(a)$ for sufficiently large odd $n$. Our results are a generalisation of the $s=2$ case which was proven in \cite{bar2011index}.
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