Fractional list packing for layered graphs

Abstract

The fractional list packing number $χ_{\ell}^{\bullet}(G)$ of a graph $G$ is a graph invariant that has recently arisen from the study of disjoint list-colourings. It measures how large the lists of a list-assignment $L:V(G)\rightarrow 2^{\mathbb{N}}$ need to be to ensure the existence of a `perfectly balanced' probability distribution on proper $L$-colourings, i.e., such that at every vertex $v$, every colour appears with equal probability $1/|L(v)|$. In this work we give various bounds on $χ_{\ell}^{\bullet}(G)$, which admit strengthenings for correspondence and local-degree versions. As a corollary, we improve theorems on the related notion of flexible list colouring. In particular we study Cartesian products and $d$-degenerate graphs, and we prove that $χ_{\ell}^{\bullet}(G)$ is bounded from above by the pathwidth of $G$ plus one. The correspondence analogue of the latter is false for treewidth instead of pathwidth.

Figures (5)

• Figure 1: Local extension of six correspondence-colourings of a $2$-tree, from a balanced edge $ab$ to $d$, such that $ad$ or $bd$ becomes balanced as well. The bold edges indicate full identity matchings in the cover.
• Figure 2: A graph with pathwidth $3$.
• Figure 3: A $2$-fold correspondence-cover $(H,L)$ of the hypercube $Q_3$, demonstrating that $\chi_{c}^{\bullet}(Q_3) >3$. For each vertex $v$ of $Q_3$, its list $L(v)=\{1_v,2_v\}$ is indicated by a grey ellipse surrounding a white dot representing $1_v$ and a black dot representing $2_v$. For most edges $uv$ of $Q_3$, the matching between $L(u)$ and $L(v)$ is the 'identity matching', consisting of edges $1_u 1_v$ and $2_u 2_v$ in $H$. The exceptions are formed by a matching of $Q_3$ that consists of three special edges$uv$ for which the matching between $L(u)$ and $L(v)$ is 'crossing', i.e., consisting of $1_u 2_v$ and $2_u 1_v$. The special edges are chosen such that every cycle $C$ of $Q_3$ traverses an odd number of them. Therefore the restriction of $(H,L)$ to a cycle $C$ of $Q_3$ cannot contain any independent set on $|V(C)|$ vertices. From this it easily follows that every independent set of the $16$-vertex graph $H$ has size at most $5$. Hence the fractional chromatic number of $H$ is at least $16/5$, which is strictly larger than $3$.
• Figure 4: A graph $G$ with treewidth $3$ and $\chi_c^{\bullet}(G) =5$.
• Figure 5: A correspondence-cover $(L,H)$ of the graph $G$ in \ref{['fig:treewidth3_chicbullet5']}. The covergraph $H$ has fractional chromatic number $>4$ and all lists of size $\leqslant 4$, thus certifying that $\chi_c^{\bullet}(G) >4$. The grey ellipses indicate the lists. To avoid clutter, only the matchings on the central $K_4$ of $G$ are depicted.

Theorems & Definitions (46)

• Lemma 1.1: DNP19
• Theorem 2.1: BMS22
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4: BJKM06
• Theorem 2.5
• Theorem 2.6: KMMP22
• Theorem 2.7
• Definition 3.1
• Definition 3.2