路划分猜想成立及其对迂回色数和星色数的有效性
The Path Partition Conjecture is True and its Validity Yields Upper Bounds for Detour Chromatic Number and Star Chromatic Number
摘要：
图$G$的迂回顺序，记为$τ(G)$，是$G$中最长路径的顺序。$V(G)$的分区$(A, B)$使得$τ(\langle A \rangle) \leq a$和$τ(\langle B \rangle) \leq b$被称为$G$的$(a, b)$ -分区。一个图$G$称为$τ$ -可分图，如果$G$对每对正整数$(a, b)$都有一个$(a, b)$ -可分图，使得$a + b = τ(G)$ .著名的路径划分猜想指出，每个图都是$τ$ -可划分的。在\cite{df07}中Dunber和Frick证明了如果每个2-连通图都是$τ$ -可分的，那么每个图都是$τ$ -可分的。本文证明了2-连通图是PDMATHAMT-可分图。因此，我们的结果肯定地解决了路径划分猜想。我们证明了以下两个定理作为路划分猜想有效性的推论。{ \bf定理1：}对每个图$G$，$χ_s(G) \leq τ(G)$，其中$χ_s(G)$是图$G$的星色数.图$G$的$n^{th}$迂回色数，记作$χ_n(G)$，是给$G$的顶点着色所需的最小颜色数，使得没有阶数大于$n$的路是单色的。这些色数是由Chartrand，Gellar和Hedetniemi引入的\cite{cg68}作为点色数$χ(G)$的推广。{ \bf定理2：}对每个图$G$和每个$n \geq 1$，有$χ_n(G) \leq \left\lceil \frac{τ_n(G)}{n} \right\rceil$，其中$χ_n(G)$表示$n^{th}$迂回色数。定理2解决了Frick和Bullock的\cite{fb01}猜想：对每个图$G$，对每个$n \geq 1$，肯定地存在$χ_n(G) \leq \left\lceil \frac{τ(G)}{n} \right\rceil$.
英文摘要：
The detour order of a graph $G$, denoted $τ(G)$, is the order of a longest path in $G$. A partition $(A, B)$ of $V(G)$ such that $τ(\langle A \rangle) \leq a$ and $τ(\langle B \rangle) \leq b$ is called an $(a, b)$-partition of $G$. A graph $G$ is called $τ$-partitionable if $G$ has an $(a, b)$-partition for every pair $(a, b)$ of positive integers such that $a + b = τ(G)$. The well-known Path Partition Conjecture states that every graph is $τ$-partitionable. In \cite{df07} Dunber and Frick have shown that if every 2-connected graph is $τ$-partitionable then every graph is $τ$-partitionable. In this paper we show that every 2-connected graph is $τ$-partitionable. Thus, our result settles the Path Partition Conjecture affirmatively. We prove the following two theorems as the implications of the validity of the Path Partition Conjecture.\\ {\bf Theorem 1:} For every graph $G$, $χ_s(G) \leq τ(G)$, where $χ_s(G)$ is the star chromatic number of a graph $G$. The $n^{th}$ detour chromatic number of a graph $G$, denoted $χ_n(G)$, is the minimum number of colours required for colouring the vertices of $G$ such that no path of order greater than $n$ is mono coloured. These chromatic numbers were introduced by Chartrand, Gellar and Hedetniemi\cite{cg68} as a generalization of vertex chromatic number $χ(G)$.\\ {\bf Theorem 2:} For every graph $G$ and for every $n \geq 1$, $χ_n(G) \leq \left\lceil \frac{τ_n(G)}{n} \right\rceil$, where $χ_n(G)$ denote the $n^{th}$ detour chromatic number.\\ Theorem 2 settles the conjecture of Frick and Bullock \cite{fb01} that $χ_n(G) \leq \left\lceil \frac{τ(G)}{n} \right\rceil$, for every graph $G$, for every $n \geq 1$, affirmatively.
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