Let $P_k$ be a path on $k$ vertices. In an earlier paper we have proved that each polyhedral map $G$ on any compact $2$-manifold $\mathbb{M}$ with Euler characteristic $\chi (\mathbb{M})\le 0$ contains a path $P_k$ such that each vertex of this path has, in $G$, degree $\le k\left\lfloor \frac{5+\sqrt{49-24 \chi (\mathbb{M})}}{2}\right\rfloor $. Moreover, this bound is attained for $k=1$ or $k\ge 2$, $k$ even. In this paper we prove that for each odd $k\ge \frac{4}{3} \left\lfloor \frac{5+\sqrt{49-24\chi (\mathbb{M})}}{2}\right\rfloor +1$, this bound is the best possible on infinitely many compact $2$-manifolds, but on infinitely many other compact $2$-manifolds the upper bound can be lowered to $\left\lfloor (k-\frac{1}{3})\frac{5+\sqrt{49-24\chi (\mathbb{M})}}{2}\right\rfloor $.
The well known types of routes in graphs and directed graphs, such as walks, trails, paths, and induced paths, are characterized using axioms on vertex sequences. Thus non-graphic characterizations of the various types of routes are obtained.
By a ternary system we mean an ordered pair $(W, R)$, where $W$ is a finite nonempty set and $R \subseteq W \times W \times W$. By a signpost system we mean a ternary system $(W, R)$ satisfying the following conditions for all $x, y, z \in W$: if $(x, y, z) \in R$, then $(y, x, x) \in R$ and $(y, x, z) \notin R$; if $x \ne y$, then there exists $t \in W$ such that $(x, t, y) \in R$. In this paper, a signpost system is used as a common description of a connected graph and a spanning tree of the graph. By a ct-pair we mean an ordered pair $(G, T)$, where $G$ is a connected graph and $T$ is a spanning tree of $G$. If $(G, T)$ is a ct-pair, then by the guide to $(G,T)$ we mean the ternary system $(W, R)$, where $W = V(G)$ and the following condition holds for all $u, v, w \in W$: $(u, v, w) \in R$ if and only if $uv \in E(G)$ and $v$ belongs to the $u-w$ path in $T$. By Proposition 1, the guide to a ct-pair is a signpost system. We say that a signpost system is tree-controlled if it satisfies a certain set of four axioms (these axioms could be formulated in a language of the first-order logic). Consider the mapping $\phi $ from the class of all ct-pairs into the class of all signpost systems such that $\phi ((G, T))$ is the guide to $(G, T)$ for every ct-pair $(G, T)$. It is proved in this paper that $\phi $ is a bijective mapping from the class of all ct-pairs onto the class of all tree-controlled signpost systems.
In this paper, by a travel groupoid is meant an ordered pair $(V, *)$ such that $V$ is a nonempty set and $*$ is a binary operation on $V$ satisfying the following two conditions for all $u, v \in V$: \[ (u * v) * u = u; \text{ if }(u * v ) * v = u, \text{ then } u = v. \] Let $(V, *)$ be a travel groupoid. It is easy to show that if $x, y \in V$, then $x * y = y$ if and only if $y * x = x$. We say that $(V, *)$ is on a (finite or infinite) graph $G$ if $V(G) = V$ and \[ E(G) = \lbrace \lbrace u, v\rbrace \: u, v \in V \text{ and } u \ne u * v = v\rbrace . \] Clearly, every travel groupoid is on exactly one graph. In this paper, some properties of travel groupoids on graphs are studied.