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### On the distance function of a connected graph

Czechoslovak Mathematical Journal

An axiomatic characterization of the distance function of a connected graph is given in this note. The triangle inequality is not contained in this characterization.

### New proof of a characterization of geodetic graphs

Czechoslovak Mathematical Journal

In [3], the present author used a binary operation as a tool for characterizing geodetic graphs. In this paper a new proof of the main result of the paper cited above is presented. The new proof is shorter and simpler.

### A new proof of a characterization of the set of all geodesics in a connected graph

Czechoslovak Mathematical Journal

### A theorem for an axiomatic approach to metric properties of graphs

Czechoslovak Mathematical Journal

### Geodesics and steps in a connected graph

Czechoslovak Mathematical Journal

### The hamiltonian chromatic number of a connected graph without large hamiltonian-connected subgraphs

Czechoslovak Mathematical Journal

If $G$ is a connected graph of order $n\ge 1$, then by a hamiltonian coloring of $G$ we mean a mapping $c$ of $V\left(G\right)$ into the set of all positive integers such that $|c\left(x\right)-c\left(y\right)|\ge n-1-{D}_{G}\left(x,y\right)$ (where ${D}_{G}\left(x,y\right)$ denotes the length of a longest $x-y$ path in $G$) for all distinct $x,y\in V\left(G\right)$. Let $G$ be a connected graph. By the hamiltonian chromatic number of $G$ we mean $min\left(max\left(c\left(z\right);\phantom{\rule{0.166667em}{0ex}}z\in V\left(G\right)\right)\right),$ where the minimum is taken over all hamiltonian colorings $c$ of $G$. The main result of this paper can be formulated as follows: Let $G$ be a connected graph of order $n\ge 3$. Assume that there exists a subgraph...

### On properties of a graph that depend on its distance function

Czechoslovak Mathematical Journal

If $G$ is a connected graph with distance function $d$, then by a step in $G$ is meant an ordered triple $\left(u,x,v\right)$ of vertices of $G$ such that $d\left(u,x\right)=1$ and $d\left(u,v\right)=d\left(x,v\right)+1$. A characterization of the set of all steps in a connected graph was published by the present author in 1997. In Section 1 of this paper, a new and shorter proof of that characterization is presented. A stronger result for a certain type of connected graphs is proved in Section 2.

### Travel groupoids

Czechoslovak Mathematical Journal

In this paper, by a travel groupoid is meant an ordered pair $\left(V,*\right)$ such that $V$ is a nonempty set and $*$ is a binary operation on $V$ satisfying the following two conditions for all $u,v\in V$: $\left(u*v\right)*u=u;\phantom{\rule{4.0pt}{0ex}}\text{if}\phantom{\rule{4.0pt}{0ex}}\left(u*v\right)*v=u,\phantom{\rule{4.0pt}{0ex}}\text{then}\phantom{\rule{4.0pt}{0ex}}u=v.$ Let $\left(V,*\right)$ 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 $\left(V,*\right)$ is on a (finite or infinite) graph $G$ if $V\left(G\right)=V$ and $E\left(G\right)=\left\{\left\{u,v\right\}\phantom{\rule{0.222222em}{0ex}}u,v\in V\phantom{\rule{4.0pt}{0ex}}\text{and}\phantom{\rule{4.0pt}{0ex}}u\ne u*v=v\right\}.$ Clearly, every travel groupoid is on exactly one graph. In this paper, some properties of travel groupoids on graphs are studied.

### Characterizing the maximum genus of a connected graph

Czechoslovak Mathematical Journal

### Signpost systems and spanning trees of graphs

Czechoslovak Mathematical Journal

By a ternary system we mean an ordered pair $\left(W,R\right)$, where $W$ is a finite nonempty set and $R\subseteq W×W×W$. By a signpost system we mean a ternary system $\left(W,R\right)$ satisfying the following conditions for all $x,y,z\in W$: if $\left(x,y,z\right)\in R$, then $\left(y,x,x\right)\in R$ and $\left(y,x,z\right)\notin R$; if $x\ne y$, then there exists $t\in W$ such that $\left(x,t,y\right)\in R$. 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 $\left(G,T\right)$, where $G$ is a connected graph and $T$ is a spanning tree of $G$. If $\left(G,T\right)$ is a ct-pair, then by the guide to...

### On a certain numbering of the vertices of a hypergraph

Czechoslovak Mathematical Journal

### A generalization of Hamiltonian cycles for trees

Czechoslovak Mathematical Journal

### On 2-factors in squares of graphs

Czechoslovak Mathematical Journal

### On the existence of 1-factors in partial squares of graphs

Czechoslovak Mathematical Journal

### Hypergraphs and intervals

Czechoslovak Mathematical Journal

### A note on upper embeddable graphs

Czechoslovak Mathematical Journal

### On locally quasiconnected graphs and their upper embeddability

Czechoslovak Mathematical Journal

### Upper embeddable factorizations of graphs

Czechoslovak Mathematical Journal

### A new characterization of the maximum genus of a graph

Czechoslovak Mathematical Journal

### An axiomatic approach to metric properties of connected graphs

Czechoslovak Mathematical Journal

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