Solution to a problem of Katona on counting cliques of weighted graphs
Peter Borg, Carl Feghali, Rémi Pellerin
TL;DR
This work analyzes Katona's question on minimizing the $k$-clique count $\pi_k(G(w))$ in blow-up graphs $G(w)$ under $m$-weightings, introducing uniform-$\alpha$ weightings and the Sperner graph $B_n$. Using a weight-shifting framework and a novel multi-edge shift lemma, the authors show that for certain graph classes—including Sperner graphs, complete multipartite graphs, and chordal graphs—the minimum $\pi_k(G(w))$ is attained when weights are uniform on a largest independent set, extending and unifying prior results for $k=2$. However, they demonstrate that this minimization principle fails for some graphs when $k\ge3$, via explicit counterexamples (notably involving $B_n$) and Bohman’s triangle-free graphs, indicating a sharp dichotomy depending on graph structure. The paper also discusses related conjectures and the limits of the uniform-$\alpha$ strategy, highlighting both the reach and the boundaries of current methods in weighted clique counting within graph blow-ups.
Abstract
A subset $I$ of the vertex set $V(G)$ of a graph $G$ is called a $k$-clique independent set of $G$ if no $k$ vertices in $I$ form a $k$-clique of $G$. An independent set is a $2$-clique independent set. Let $π_k(G)$ denote the number of $k$-cliques of $G$. For a function $w: V(G) \rightarrow \{0, 1, 2, \dots\}$, let $G(w)$ be the graph obtained from $G$ by replacing each vertex $v$ by a $w(v)$-clique $K^v$ and making each vertex of $K^u$ adjacent to each vertex of $K^v$ for each edge $\{u,v\}$ of $G$. For an integer $m \geq 1$, consider any $w$ with $\sum_{v \in V(G)} w(v) = m$. For $U \subseteq V(G)$, we say that $w$ is uniform on $U$ if $w(v) = 0$ for each $v \in V(G) \setminus U$ and, for each $u \in U$, $w(u) = \left\lfloor m/|U| \right\rfloor$ or $w(u) = \left\lceil m/|U| \right\rceil$. Katona asked if $π_k(G(w))$ is smallest when $w$ is uniform on a largest $k$-clique independent set of $G$. He placed particular emphasis on the Sperner graph $B_n$, given by $V(B_n) = \{X \colon X \subseteq \{1, \dots, n\}\}$ and $E(B_n) = \{\{X,Y\} \colon X \subsetneq Y \in V(B_n)\}$. He provided an affirmative answer for $k = 2$ (and any $G$). We determine graphs for which the answer is negative for every $k \geq 3$. These include $B_n$ for $n \geq 2$. Generalizing Sperner's Theorem and a recent result of Qian, Engel and Xu, we show that $π_k(B_n(w))$ is smallest when $w$ is uniform on a largest independent set of $B_n$. We also show that the same holds for complete multipartite graphs and chordal graphs. We show that this is not true of every graph, using a deep result of Bohman on triangle-free graphs.