Four Color Theorem (4CT) states that every planar graph is four colorable. There are two proofs given by [Appel,Haken 1976] and [Robertson,Sanders,Seymour,Thomas 1997]. Both these proofs are computer-assisted and quite intimidating.
There are several conjectures in graph theory that imply 4CT. Resolution of these conjectures probably requires a better understanding of the proofs of 4CT. Here is one such conjecture :
Conjecture : Let $G$ be a planar graph, let $C$ be a set of colors and $f : C \rightarrow C$ a fixed-point free involution. Let $L = (L_v : v \in V(G))$ be such that
- $|L_v| \geq 4$ for all $v \in V$ and
- if $\alpha \in L_v$ then $f(\alpha) \in L_v$ for all $v \in V$, for all $\alpha \in C$.
Then there exists an $L$-coloring of the graph $G$.
If you know such conjectures implying 4CT, please list them one in each answer. I could not find a comprehensive list of such conjectures.