# Criteria to Identify Cycles in the Set of Returning Paths

Let's consider a cubic bipartite graph $$G$$ with a $$3$$-edge coloring (label the colors $$-1,0,1$$) and further, paths $$p\in P$$ on $$G$$ without backtracking of length $$|p|$$, that return to the origin.

The adjacency matrix $$A$$ of $$G$$ can be split in three due to the edge coloring: $$A=A_{-1}+A_0+A_1$$ Starting from a initial vertex $$v_0$$, paths without backtracking can be written as a sequence of subsequent matrix multiplication of $$A_k$$ with $$k\in\{-1,0,1\}$$ and $$A_mA_l\neq A^2_m$$, e.g.: $$A_0A_1A_0A_{-1} ... A_1A_0v_0$$ and if we think of $$A_kv_0$$ being an initial edge, we recognize that our non-backtracking path, will necessarily continue with an edge $$A_{k{\color{red} \pm} 1 \bmod 3}$$. So we finally store our path in a sequence $$^p\Delta$$ of $$(|p|-1)$$ "$${\color{red} \pm}$$"'s., e.g. $$(+,...,+,+,-)$$. All possible paths $$p$$ have a correponding sequence $$^p\Delta$$.

1. If a path $$p$$ is returning, the path $$q$$, with $$^{q}\Delta=-\left(^{p^{-1}}\Delta\right)$$ which is the sign-inverted, position-reversed sequence is also returning, which also holds true for cycles.

2. For simple cycles (not a concatenation of several ones), I think I found that $$\displaystyle\sum_{k=1}^{|p|-1} ({^p\Delta}) _k\bmod 3 \neq 0$$, which does not hold for other returning paths, like cycles with a tail.

Two examples:

1. a $$4$$-cycle with a sequence of coloured adajacency matrices $$A_1A_0A_{-1}A_0$$ results in $$^p\Delta=\left(0-1,-1-0,0-(-1)\right)=(-,-,+)$$, which sums up to $$1 \bmod 3$$.

2. a $$6$$-cycle with a sequence of coloured adajacency matrices $$A_1A_0A_{-1}A_1A_0A_{-1}$$ results in $$^p\Delta=(-,-,-,-,-)$$, which sums up to $$-1 \bmod 3$$.

Are there criteria for $$^p\Delta$$ to identify as well concatenated cycles in the set of paths that return to the origin?

Also other criteria for simple cycles are welcome...

Another thing to note (related to point 2) is that the sum is always equivalent to $$1 \bmod 2$$ for returning paths. So then, you have that the sum is always $$\pm 1 \bmod 6$$ for simple cycles.
Since the graph is bipartite, any returning path $$p$$ must contain an even number of edges. This means that we have an odd number of elements in the sequence $${^p}\Delta$$. If we take the sum of the $$p-1$$ elements in the sequence modulo 2, we get $$(p-1) \times 1 \equiv 1$$.