A Brooks-type theorem for the k-choosability of graphs with maximum local edge-connectivity k

Abstract

For a graph $G$ with at least two vertices, the maximum local edge-connectivity of $G$ is the maximum number of edge-disjoint $(u,v)$-paths over all distinct pairs of vertices $(u,v)$ in $G$. Stiebitz and Toft (2018) proved a Brooks-type theorem for graphs with maximum local edge-connectivity $k$, showing that a graph with maximum local edge-connectivity $k$ is not $k$-colourable if and only if it has a block in $\mathcal{H}_k$, which is the class of graphs that can be obtained by taking Hajós joins of copies of $K_{k+1}$ and, when $k=3$, odd wheels. We prove that a $2$-connected graph with maximum local edge-connectivity $k$ is $k$-choosable if and only if it is not in $\mathcal{H}_k$. On the other hand, deciding $k$-choosability when restricted to graphs with maximum local edge-connectivity $k$ (that might not be $2$-connected) is $Π_2$-complete. To prove the former result, we first prove several generalisations of a well-known characterisation of degree-choosability; these may be of independent interest.

Figures (16)

• Figure 1: A Hajós join
• Figure 2: A graph $H$ with maximum local edge-connectivity $3$ that is not $3$-choosable (illustrated by the particular $3$-list assignment), even though $H$ is $3$-colourable, and each block of $H$ is $3$-choosable.
• Figure 3: Polar graphs that are not choosable, but the underlying graph is choosable.
• Figure 4: An example of a colour deletion. At the top is a polar graph with a polar assignment, and we consider a colouring $\phi$ where the central vertex $v$ is coloured red. At the bottom is the resulting $(\{v\},\phi)$-colour deletion.
• Figure 5: An example of a polar assignment union of two copies of a polar graph we call $T_3$ (with no polarised edges), that have a single vertex in common, shown in red.

Theorems & Definitions (212)

• Theorem 1.1: Vizing Vizing1976; Erdős, Rubin, and Taylor ERT1979
• Theorem 1.2: Stiebitz and Toft ST2018
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.4: Vizing Vizing1976; Erdős, Rubin, and Taylor ERT1979
• Proposition 2.5