30 problems

Subcritical concentration conjecture. The size of the largest component is concentrated within a factor $1+o_{\rm p}(1)$ also in the subcritical case.

Wormald's conjecture. The random $d$-regular graph and the random $d$-process cannot be distinguished with high confidence.

Let a graph have $N$ vertices, and construct a configuration-model graph by assigning each vertex a prescribed number of stubs and pairing the stubs uniformly at random, allowing s…

Magnetization-threshold conjecture. There exists $\tilde\alpha>0$, depending only on $\beta$ and $\nu$, such that $o$ is asymptotically magnetized if

Asymptotic tightness conjecture. Supported by Riordan's result for constant maximum degree, the upper bound in $$ is asymptotically tight for all degree sequences that satisfy some…

Hofstad–Janson–Łuczak conjecture. $L_1({{\mathbb{CM} }}_{n})$ is concentrated in the barely subcritical regime.

Let $D\in\mathbb{N}$ with $D\geq 3$, and suppose that $p_k=0$ and $n_k=0$ for $k\ne D$, while $p_D=1$ and $n_D=n$. For each $x\in[0,1]$, let $q_D^{(x)}=x$, let $q_k^{(x)}=0$ for…

Let $D\in\mathbb{N}$ with $D\geq 3$, and suppose that $p_k=0$ and $n_k=0$ for $k\ne D$, while $p_D=1$ and $n_D=n$. Fix $M\in\mathbb{N}$ and $q_D^{(i)}\in(0,1]$ for $i=1,\dotsc,M$,…

Let ${\mathbf{d}}$ be a graphic degree sequence, and construct the switched configuration model by repeatedly choosing a bad edge—one that is a loop or parallel to another edge—and…

The paper studies the degree-greedy algorithm for finding independent sets in configuration-model graphs, under asymptotic degree distributions satisfying the hypotheses of Theorem…

Let $r_n$ be the Pearson assortativity coefficient of the configuration model in the scale-free regime $1<\gamma<2$. Scaling Pearson conjecture. As $n\to\infty$, there exists a ran…

Let ${\bf D}_n$ be sampled from $\mathscr{D}$ with $1<\gamma<2$, and let $\widehat{G}_n=\operatorname{ECM}({\bf D}_n)$. Let $Z_n$ denote the number of removed edges in the erased c…

Let the erased configuration model (ECM) be the random graph obtained from the configuration model by removing self-loops and replacing multiple edges by single edges. Let a unifor…

Consider competing first passage percolation on a configuration model with finite-variance degrees and exponential edge weights, with infection types of intensities $\lambda_1$ and…

Let $P$ and $Q$ be the vertex and edge-type probability distributions in the approximate assortative configuration graph construction. For $E$ matching edges, let…

Slower-color size conjecture. The number of vertices painted by the slower color should agree with the result of BarHofKom14 whenever these fluctuations are tight. This refines the…

Let ${\mathrm{CM}}_n(\boldsymbol{d})$ be a configuration model with power-law exponent $\tau\in(2,3)$, and suppose edge weights have the form $c+X$ for a constant $c>0$ and a rando…

In the setting of uniformly chosen source vertices in ${\mathrm{CM}}_n(\boldsymbol{d})$ with power-law exponent $\tau\in(2,3)$, suppose the branching processes associated with the…

Let ${\mathrm{CM}}_n(\boldsymbol{d})$ have i.i.d. power-law degrees with exponent $\tau\in(2,3)$, let $D^\star$ be the size-biased offspring distribution, and let $I_r$ and $I_b$ b…

Let ${\mathrm{CM}}_n(\boldsymbol{d})$ be a configuration model with i.i.d. degrees whose distribution has power-law exponent $\tau$, and assign i.i.d. continuous edge weights. Fini…

Let $Y_r^{\scriptscriptstyle{(n)}}$ and $Y_b^{\scriptscriptstyle{(n)}}$ denote the limiting growth variables for the red and blue processes, and let $\mathcal{B}_\infty$ be the eve…

Noncoexistence conjecture for two explosive processes. If the spreading dynamics are such that both underlying branching processes are explosive, then there is never coexistence an…

Sublinear polynomial loser conjecture. There is still no coexistence with high probability, and the loser type can paint a polynomial number of vertices with a random exponent that…

Consider competing first passage percolation on the configuration model in the finite-variance regime $\tau>3$, and compare it with the random regular-graph case, where the degree…

Let $\widetilde{RCM}[n]$ be the weighted repeated configuration model on $n$ vertices obtained from a given degree sequence, with independent identically distributed edge lengths.…