4 problems

Let $G$ be a graph, let $[200~\lambda(G)$ denote its spectral radius, let $c_G(v)$ be the maximum order of a clique in $G$ containing $v$, and let $w_r(v)$ be the number of walks o…

Let $G$ be a nice graph of size $m$, and let ${\rm ML}^{\rm W}(G)$ denote the minimum size of an irregularising walk of $G$. The 3m conjecture for irregularising walks states that…

A double-tandem walk is a walk for which the starting point is $(x_{start},y_{start})$, and $x_{min}$ and $y_{min}$ denote the minimal $x$- and $y$-coordinates visited during the w…

Let the slit plane be the integer lattice $\mathbb Z^2$ with the half-axis $\{(k,0):k\le 0\}$ removed. Consider paths starting at $(1,0)$ that use $2n+1$ horizontal or vertical uni…