A proper coloring of the vertices of a graph $G$ is seen as a homomorphism from the graph vertices to the complete graph on the number of vertices equal to the chromatic number of the graph. Similarly, the fractional coloring or $a:b$-coloring of vertices is seen as a homomorphism from the graph vertices to a Kneser graph $K(a,b)$ that is a graph whose vertices are $b$ subsets of an $a$-set with two vertices adjacent iff they are disjoint.

I wish to ask if there is a similar homomorphism characterization of the list coloring. I think this is somewhat related to the recent area of DP-coloring or correspondence coloring as in the paper. Any hints in this regard. Thanks beforehand.


1 Answer 1


If I googled 'list coloring' correctly, it is the following: For a graph $G$ we have at every vertex a given subset of colors. A lsit coloring chooses at every vertex one color from that list so that two adjacent vertices always have a different color.

From these lists we can form a new graph $G'$ with vertices pairs $(v,c)$ where $v$ is a vertex of $G$ and $c$ is a viable color of $v$. Two vertices $(v,c),(v',c')$ are adjacent, if $v$ and $v'$ are adjacent and $c\neq c'$.

We then have two canonical maps $G'\to G$ and $G'\to C_n$, where $C_n$ is the complete graph with $n$ vertices (given by the $n$ colors).

In this language a list coloring is a section of the map $G'\to G$.

  • $\begingroup$ For example if all colors are allowed everywhere, then $G'=G\times C_n$ and hence a section $G\times C_n\rightarrow G$ is just a map $G\rightarrow C_n$. $\endgroup$ Commented Apr 4, 2022 at 13:40
  • $\begingroup$ In the OP's question, a homomorphism FROM the graph to either the complete graph (for colourings) or the Kneser graph (for fractional colourings) determines a valid colouring (or fractional colouring). So the "graph-to-be-coloured" is the domain of the homomorphism and the set of colours (encoded in some fashion) is the codomain. But here you are talking about a homomorphism TO the graph, and I can't see how this actually determines a list-colouring. $\endgroup$ Commented Apr 5, 2022 at 5:28
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    $\begingroup$ A section of $p:G'→G$ is a map $s:G→G′$ such that $p\circ s$ is the identity. So it is still a map from G to somewhere. The coloring of g is given by picking the second coordinate of s(g). $\endgroup$ Commented Apr 5, 2022 at 9:33
  • $\begingroup$ I am sorry if I am being dense here, but if I am correct, for each vertex $v \in G$ there is a set $(v,c_1)$, $(v,c_2)$, etc. of vertices of $G'$ with one vertex for each colour in the list for $v$, call this set a fibre. Then the map $G'\rightarrow G$ is just the projection onto the first coordinate. A section is then a choice of one vertex from each fibre. But how does this encode the colouring restriction? It seems like we can't take any section, but need the section ($s$ in your notation) to actually be a homomorphism? $\endgroup$ Commented Apr 6, 2022 at 1:37
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    $\begingroup$ Say we have two adjacent vertices $g,g'$ and $s(g)=(g,v)$ and $s(g')=(g',v')$, Since $s$ is a morphism in the category of graphs, we have that $g,g'$ are connected and $v,v'\in C_n$ are connected, i.e. $v,v'$ are different. $\endgroup$ Commented Apr 6, 2022 at 9:32

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