We prove that a path maximizes the expected range of a uniformly chosen graph homomorphism into the integers, with one vertex pinned at zero, among all connected bipartite graphs of the same order. This establishes the expectation form of the Benjamini--H\"aggstr\"om--Mossel conjecture. The proof restricts and rescales a homomorphism on each bipartition class, then contracts the edges on which the resulting height function is constant. A quantitative estimate for the rank of these zero edges compensates for a parity term in the expected range of a simple random walk, allowing an induction on the number of vertices. We then prove that the BHM inequality implies the Loebl--Ne\v set\v ril--Reed inequality for uniformly chosen integer 1-Lipschitz functions on arbitrary connected graphs, and hence obtain the LNR conjecture as a corollary of BHM. The proof was obtained through interaction with OpenAI GPT-6 Astra and verified by the author. The main results have also been formalized and checked in Lean~4.
Inspect the vertices of a finite simple graph in uniformly random order, accepting each vertex if none of its neighbors has previously been accepted. Let $X_G$ be the number of accepted vertices. For every triangle-free graph with $n\ge2$ vertices and $e$ edges, we prove $\operatorname{Var}(X_G)\le e((n-2)/n)^2$, with...
For positive integers with , a circular ‐flow of a graph is defined as a pair where represents an orientation of , and is a function mapping the edges of to the set such that, at each vertex, the sum of the ‐values of the incoming edges equals the sum of the ‐values of the outgoing edges. The flow index of , denoted...
Jia-Ao Li, Xin-Yuan Li· Journal of Graph Theory· 0 citations
We establish sharp quadratic bounds on the relaxation time of simple random walk on connected regular graphs, with leading constants $3/(2\pi^2)$ and $1/\pi^2$ for even and odd orders, respectively. This resolves a longstanding conjecture known as Aldous--Fill spectral gap conjecture (2002). We also prove uniqueness an...
We prove an annealed central limit theorem for the weight of the maximum weight matching on uniformly random simple graphs with prescribed, uniformly bounded degrees and i.i.d. exponential edge weights. In particular, the result applies to random $d$-regular graphs for every fixed $d \ge 2$. The proof separates the flu...
The representation number of a graph is the least positive integer $k$ for which its vertices can be arranged in a word, each occurring $k$ times, so that two distinct letters alternate precisely when the corresponding vertices are adjacent. We prove that every bipartite graph on $N\ge9$ vertices has representation num...
Matthew J. Colbrook, Catherine Drysdale· 0 citations
We consider site percolation on general graphs and give a sufficient condition on the escape probability of a certain random walk with degree-dependent conductances, such that the number of minimal vertex cutsets of size $n$ separating a given vertex from infinity is bounded above exponentially in $n$. This result exte...
Joel Bassil· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
