# Cycle decompositions of locally linear 4-regular graphs

(Preface)
We consider only finite, simple, undirected graphs here. An orientation of a graph $$G$$ is obtained by assigning some direction to each edge of $$G$$.

(Question starts)
A graph is locally linear if every edge is in exactly one triangle. Let $$G$$ be a connected locally linear 4-regular graph with an orientation $$\overrightarrow{G}$$ which orients every triangle $$(u,v,w)$$ in $$G$$ cyclically (i.e., either $$u\to v\to w$$ or $$v\to u\to w$$).
[Side note: $$\overrightarrow{G}$$ is an Eulerian orientation in the sense that every vertex has two in-coming edges and two out-going edges.]
A direction alternating cycle in $$\overrightarrow{G}$$ is a cycle $$(v_1,v_2,\dots,v_k)$$ such that direction of adjacent edges alternate (that is, either $$v_1\to v_2\leftarrow v_3\to v_4\leftarrow \dots v_k\leftarrow v_1$$ or $$v_1\leftarrow v_2\to v_3\leftarrow v_4\to \dots v_k\to v_1$$). Note that in a direction alternating cycle in $$\overrightarrow{G}$$, adjacent edges in the cycle belong to different triangles in $$G$$ (becuase triangles are oriented cyclically); that is for each direction alternating cycle $$D_i$$ in $$G$$, $$G[V(D_i)]$$ is trianngle-free. Clearly, $$\overrightarrow{G}$$ can be (edge) decomposed into direction alternating cycles.

Problem 1: Characterize orientations $$\overrightarrow{G}$$ that admit a cycle decomposition into exactly four direction alternating cycles.

Since a complete characterization may be challenging, let us consider necessary/sufficient conditions.
It seems that $$G$$ should be 4-connected (there exist graphs $$G$$ with vertex connectivity 3 such that some orientation $$\overrightarrow{G}$$ of $$G$$ does not decompose into four direction alternating cycles).
For a concrete special case, assume that $$G$$ is also 4-connected and planar, and $$\overrightarrow{G}$$ orients every face boundary cyclically.
Problem 2: Can we say that $$\overrightarrow{G}$$ admits a cycle decomposition into exactly four direction alternating cycles?

It seems that line graph of $$CL_{4q}$$ has such a decomposition. $$CL_{4q}$$ is the circular ladder graph such that inner and outer cycles are the cycle $$C_{4q}$$; e.g.: $$CL_4$$ is the cube (also called 3-cube). Such decompositions of $$L(CL_4)$$ and $$L(CL_8)$$ are given below (all intersection points are vertices, and the four cycles are given distinct colours). Cycle decompositions $$D=\{D_1,D_2,\dots,D_t\}$$ of $$G$$ such that $$G[V(D_i)]$$ is triangle-free for $$1\leq i\leq t$$ are in one-to-one correspondence with cycle double covers of clique graph $$K(G)$$ of $$G$$ (it is known that $$G$$ will be a cubic graph and line of graph of $$K(G)$$ will be $$G$$).

PS: Unfortunately, graphclasses.org does not include the class locally linear. Any reference that deal with decompositions of locally linear graphs is welcome; hence the tag reference-request. Of course, such graphs admit a triangle decomposition, a $$P_3$$-decomposition, and a $$P_4$$-decomposition (the last one is because #edges is a multiple of three).