The colorings of homeomorphisms on connected graphs (Q409600)

For a map \(f\) from a space \(X\) to itself, a cover \(\mathcal U\) of \(X\) is a \textit{coloring} of \((X,f)\) if \(f(U)\cap U=\emptyset\) for each \(U\in{\mathcal U}\), and the \textit{color number} col\((X,f)\) is defined by \(\min\{|{\mathcal U}|:{\mathcal U}\text{ is a coloring of }(X,f)\}\). Note that col\((X,f)\) may not be finite. For an upper bound of the color number, the following results were obtained: Let \(X\) be a paracompact Hausdorff space with \(\dim X\leq n\). If \(f:X\to X\) is a fixed-point free homeomorphism, i.e. \(f(x)\neq x\) for each \(x\in X\), then col\((X,f)\leq n+3\). In addition, if \(f^2(x)=x\) for each \(x\in X\), then col\((X,f)\leq n+2\). On the other hand, if a space \(X\) is connected and \(f:X\to X\) is fixed-point free homeomorphism, then col\((X,f)\geq3\).NEWLINENEWLINEFrom the above results, the following natural question appeared in [\textit{A. Krawczyk} and \textit{J. Steprāns}, Topology Appl. 51, No. 1, 13--26 (1993; Zbl 0838.54025)] and [\textit{Y. Akaike, N. Chinen} and \textit{K. Tomoyasu}, Bull. Pol. Acad. Sci., Math. 57, No. 1, 63--74 (2009; Zbl 1180.54048)]: Let \(X\) be a 1-dimensional connected space and \(f:X\to X\) a fixed-point free homeomorphism. Which is true, col\((X,f)=3\) or col\((X,f)=4\)? In particular, if \(X\) is a connected finite graph, which is true, col\((X,f)=3\) or col\((X,f)=4\)?NEWLINENEWLINEIn this paper, the author answers to the latter part of the question, indeed he proves that if \(f:X\to X\) is a fixed-point free homeomorphism from a connected locally finite graph \(X\) to itself, then gcd(Per\((f))\in\{1,3\}\) if and only if col\((X,f)=4\), where gcd(Per\((f))\) means the greatest common divisor of all periods for the map \(f\). Since it is known that col\((X,f)=3\) if \(X\) is a finite graph and Per\((f)\) is nonempty satisfying gcd(Per\((f))\not\in\{1,3\}\), to obtain the above result, he proves the following: (1) If gcd(Per\((f))\in\{1,3\}\), then col\((X,f)=4\); (2) If Per\((f)= \emptyset\), then col\((X,f)=3\); (3) If \(X\) is an infinite graph and Per\((f)\) is nonempty satisfying gcd(Per\((f))\not\in\{1,3\}\), then col\((X,f)=3\).