Distel, Gollin, Harvey, Hendrey, Hickingbotham, Mohar and Wood (2023) conjectured that graphs of degree-$d$ polynomial growth can be embedded into the strong product of $d$ trees, each with linear growth, and a constant-size complete graph. Very recently, the case $d = 4$ of the conjecture was disproved by Illingworth, Norin and Steiner (2026). In this paper, we provide counterexamples to the conjecture for every integer $d \geq 2$, thus leaving $d=1$ as the only open case. Our counterexamples are appropriately subdivided cubic expanders. Our main contribution is to construct, for every real number $d>1$, subdivisions of cubic expanders with degree-$d$ polynomial growth and whose balanced separators have size $\Omega(n^{1-1/d}\log n)$, where $n$ denotes the number of vertices.
Bentert, Fomin, Golovach, Korhonen, Lochet, Panolan, Ramanujan, Saurabh, and Simonov (SODA 2025) initiated the parameterized study of Edge-Disjoint Shortest Cycle Packing: given a weighted graph $G$ and an integer $k$, decide whether $G$ contains $k$ edge-disjoint cycles of minimum weight. They showed that the problem...
A. Agrawal, Fedor V. Fomin, P. Golovach et al.· 0 citations
In 1979, Albertson and Berman conjectured that every planar graph $G$ contains an induced forest of order at least $|V(G)|/2$. This long-standing conjecture was recently disproved by several explicit counterexamples, which naturally led to several extremal and structural questions that we answer. We combine mathematica...
Wouter Cames van Batenburg, J. Goedgebeur, Jorik Jooken· 0 citations
Henning and Yeo conjectured an upper bound on the identifying vertex cover number of a graph in terms of its order, size, and maximum degree. A two-parameter family $H_{t,r}$ of connected diameter-two graphs disproves the bound for every maximum degree at least four; after denominators are cleared, its margin is exactl...
Tutte's flow conjectures ask when a graph has a nowhere-zero flow with small integer values. We approach these questions through spanning-tree packing and an analytic minimization argument of Alon, Buci\'c, and Davies. For every integer $q\ge2$, we prove that $q$ edge-disjoint spanning trees suffice to realize every co...
Aldous and Fill (2002) conjectured that the maximum relaxation time of a random walk on a connected regular graph with $n$ vertices is bounded above by $(1+o(1))\frac{3n^2}{2\pi^2}$, with asymptotic equality for even $n$. Since the relaxation time of a $d$-regular graph $G$ is $d/\mu(G)$, where $\mu(G)$ denotes its alg...
For a graph $G$, let $a(G)$ be the maximum number of vertices in an induced forest. The Albertson-Berman conjecture, posed in 1979, asserts that every $n$-vertex planar graph satisfies $a(G)\ge n/2$. Borodin's bound $a(G)\ge 2n/5$ remains the general lower bound toward this problem. We disprove the conjecture with an e...
Hee Jin Jung· 1 citation· ⚡1
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