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• W4308671133 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 $pi_k(G)$ 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 $u in U$, $w(u) = leftlfloor m/|U| rightrfloor$ or $w(u) = leftlceil m/|U| rightrceil$. Katona asked if $pi_k(G(w))$ 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 $pi_k(B_n(w))$ 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." @default.
• W4308671133 created "2022-11-14" @default.
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• W4308671133 date "2022-11-08" @default.
• W4308671133 modified "2023-09-28" @default.
• W4308671133 title "Solution to a problem of Katona on counting cliques of weighted graphs" @default.
• W4308671133 doi "https://doi.org/10.48550/arxiv.2211.04153" @default.
• W4308671133 hasPublicationYear "2022" @default.
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