10 problems

Let $G$ be a cyclically 4-connected cubic graph, and let $F(G,t)$ denote its flow polynomial. Let $\tau=(1+\sqrt{5})/2$. Finiteness conjecture. For every $\epsilon>0$, only finitel…

Let $G$ be a 3-connected graph with $n$ vertices and $m$ edges, and let $F(G,t)$ denote its flow polynomial. Let $\phi\approx1.749$ be the flow root of $K_5$ in $(1,2)$. 3-connecte…

Let $G$ be a cubic bridgeless graph with $E$ edges, let $F_G(Q)$ denote its flow polynomial, and let $\phi=(1+\sqrt{5})/2$ be the golden ratio. Golden inequality conjecture. … More…

Let $\mathtt{Frk}_{i,n-i}$ be the fork with $n$ vertices and stem of size $i$, and let $\#\mathbb{F}(\mathtt{Frk}_{i,n-i},k,0)$ denote the number of closed flows of size $k$ on thi…

Consider a family of non-planar connected graphs $G_{n,L}$ with $n$ identical layers of width $L$ and periodic boundary conditions in the $n$-direction, together with a curve…

Consider a family $G_{n,L}$ of non-planar connected graphs as above. Let $Q_\ell$ be the smallest $Q>0$ for which the transfer matrix $\widetilde{\sf T}_{L,\ell,(\ell)}$ for the fu…

For a family of generalized Petersen graphs $G(nk,k)$, let $Q_c(k)$ be the real crossing value and let $\mu_{k+1,\ell,(\ell),\star}$ denote the indicated dominant eigenvalues. Domi…

Let $G$ be a bridgeless graph, and let $\Phi_G(Q)$ denote its flow polynomial. The six-flow conjecture. For any bridgeless graph $G$, … The source presents this as a weaker conject…

Let $G$ be a bridgeless graph, and let $\Phi_G(Q)$ denote its flow polynomial. Haggard–Pearce–Royle conjecture. For any bridgeless graph $G$, … This is a proposed weakening of Wels…

Let $G$ be a bridgeless graph, and let $\Phi_G(Q)$ denote its flow polynomial. Welsh's conjecture. For any bridgeless graph $G$, … The conjecture parallels the Birkhoff–Lewis resul…