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Let $G=\left( V,E,w \right)$ is an undirected graph with $E\subseteq \mathcal{P}_{2}\left( V \right)$ where $V$ is the set of vertices, ${{\mathcal{P}}_{2}}\left( V \right)=\left\{ \alpha \subseteq V:1\le \left| \alpha \right|\le 2 \right\}$, and $w:V\to \mathbb{R}_{0}^{+}$ is a weight function with $\mathbb{R}_{0}^{+}=\left\{ r\in \mathbb{R}:r\ge 0 \right\}$. Let us define a function $W:E\to {{\mathbb{R}}_{\ge 0}}$ as follows: $$W(e)=|w(v')-w(v'')|$$

if $\left\{ {v}',{v}'' \right\} \in E$. Consider a cylce in the graph $G$ denoted by $C=\left\{ {{e}_{1}},{{e}_{2}},...,{{e}_{n}} \right\}$ and suppose that there is a $\sigma :C\to \left\{ -1,1 \right\}$ function for any $C$ such that $$\sum\limits_{i=1}^{n}{\sigma \left( {{e}_{i}} \right)W\left( {{e}_{i}} \right)}=0.$$ My question is whether all the functions $\sigma :C\to \left\{ -1,1 \right\}$ can be substituted by a „global” function $\varsigma :E\to \left\{ -1,1 \right\}$ such that for any cycle $C=\left\{ {{e}_{1}},{{e}_{2}},...,{{e}_{n}} \right\}$ the following equation holds $$\sum\limits_{i=1}^{n}{\varsigma \left( {{e}_{i}} \right)W\left( {{e}_{i}} \right)}=0.$$ Furthermore, if there is such a function, I would like to know how to construct it.

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    How does $E\subseteq \mathcal{P}(V)$ define a graph? Or do you mean that a edge $e\in \mathcal{P}(V)$ connects all the vertices $v\in e$ at once? Or did you mean $E\subseteq \mathcal{P}_2(V)$ where $\mathcal{P}_2$ denotes subset of cardinality 2? – 2017-02-21
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    Note that $\{\alpha \subseteq V: |\alpha| \le 2\}$ also allows cardinality 0 and 1, so please write $|\alpha| =2$. – 2017-02-21
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    Yes, I have allowed loops corresponding to $|\alpha|=1$. Please, focus on the problem as well. – 2017-02-21
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    Loops are no real problem, they cannot occure anyway as any positive weight will contradict the cycle-weight-property that you assume. I assume $\mathbb R^+$ excludes $0$. – 2017-02-21
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    Any graph has both properties (existance of $\sigma$ and $\xi$), by assigning the same weight to all vertices. – 2017-02-21
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    @Thorsten I think you cannot choose the vertex weights, but they are given. – 2017-02-21

2 Answers 2

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Consider the graph

a ------8------ b
| \             |
|  \            |
|    \          1
|     \         |
|      \        |
8       8       c
|        \      |
|         \     |
|          \    1
|            \  |
|             \ |
d --1-- e --1-- f

it has three cycles, each of which has a sigma (the 8's in each of the cycle must have a different signum). however there is no global $\xi$ function, because then at least two of the 8's edges would have a same sign.

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    In fact, I have a $w:V\to {{\mathbb{R}}^{+}}\cup \left\{ 0 \right\}$ function such that $W\left( e \right)=\left| w\left( {{v}'} \right)-w\left( {{v}''} \right) \right|$ for any $e=\left\{ {v}',{v}'' \right\}$. $w$ allows loops as well with zero weight. – 2017-02-21
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    @M.Winter: Thank you, I've fixed that. – 2017-02-21
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Maybe the following graph is a counterexample:

enter image description here

It is even compatible with the map $w:V\rightarrow\mathbb R^+\cup\{0\}$ you mentioned in another comment.

Explanation:

When you have a global sign function $\varsigma$ for such a graph, the sum $\sum \varsigma(e_i)W(e_i)$ over each of the three paths must be zero, otherwise it cannot cancel out pair-wise. But you cannot achieve a sum of zero for each one these paths separately.