Let $G$ be a tripartite graph with partite sets $A,B,C$. The graphs $A\cup B$, $B\cup C$ and $C\cup A$ are each bipartite. Let the maximum degree of the graph be $\Delta$.


Now, we know that the List edge coloring conjecture is true for bipartite graphs, that is, the edge choosability is same as edge chromatic number or chromatic index for bipartite graphs. Now, first we list-edge color the graph $A\cup B$, then list-edge color the graph $B\cup C$ and finally the remaining graph $C\cup A$. Then this should give a list coloring of the graph $G$ right. The lists while coloring the graphs $A\cup B$, $B\cup C$ and $C\cup A$ are are actually lists of length $\chi'(G)$, where $\chi'$ is the chromatic index. Where is the fault in this argument? Thanks beforehand.



List coloring of tripartite graph
The procedure, as such, may not give a proper coloring for a given graph. Consider the graph having $4$ vertices, $1,2,3,4$ given by the adjacency list $1-4,2-4,1-3,2-3,3-4$. Here the vertices $1,2$ form a part $A$, and vertices $3$ and $4$ belong to different parts, say $B$ and $C$ respectively. We know that $3$ is the chromatic index. Let us assume we have the list of $a,b,c$ to begin with. Consider giving the same list of $a,b$ to color the edges $1-3$ and $2-3$. We give the list colors $b,c$ to the edges $1-4$ and $2-4$. Then, we are left with a situation that we cannot color the remaining edge $3-4$.


May be, an ordering of the bipartite graphs is needed. Specifically, in the above example, if we first color the edges of the bipartite graphs incident to the vertices having largest maximum degree first, that is , first color the edges of $A\cup B$ and then color the edges of $B\cup C$, and then, finally the edges of $A\cup C$, we could arrive at a proper coloring.


