21 problems

Let ${\cal C}_{\geq k}$ denote the family of cycles of length at least $k$. For $2\leq a\leq\lfloor (k-1)/2\rfloor$, let $H(n,a,k)$ be a graph with vertex set $A\cup B\cup C$ such…

Conjecture on the structure underlying their performance. The observed performance results are a consequence of the column-weight-four array codes having minimum distance significa…

Let $d$-regular graphs have girth $g>3$, and consider the number of $k$-cycles they contain. A balanced blowup of $C_g$ is the graph obtained by replacing each vertex of the cycle…

Let $G$ be a graph with signature $s(G)$, where the signature is the difference between the numbers of positive and negative eigenvalues of its adjacency matrix. Let $c_3(G)$ and…

The $B3$ code is a quantum LDPC code whose one circulant matrix is composed of juxtaposed $6$-cycles and whose other circulant matrix has girth $10$. Girth-10 performance conjectur…

A $d_v$-$3$, $d_c$-$6$ GHP code is a code with variable-node degree $3$ and check-node degree $6$. A weight-$3$ polynomial is a polynomial having three nonzero terms, and a juxtapo…

Let $G$ be a 4-connected planar graph on $n$ vertices. A cycle of length $4$ is a cycle with four vertices. Malkevitch's conjecture. If $G$ contains a cycle of length $4$, then $G$…

Let $G$ be a 4-connected planar graph on $n$ vertices. Chen–Fan–Yu's conjecture. $G$ contains a cycle of length $\ell$ for every $\ell\in\{n,n-1,\ldots,n-25\}$ with $\ell\geq 3$. C…

Let $H_1,\ldots,H_\ell$ be the almost edge-disjoint rainbow subgraphs of excess $k$ produced by the paper's many non-star case. Short rainbow cycle conjecture for the constructed s…

Let $G$ be a planar 4-connected $n$-vertex graph, let $e$ be any edge of $G$, and let $\lambda$ satisfy … The edge $e$ is contained in cycles of $\lambda(n-2)$ pairwise distinct le…

Let $G[X,Y]$ be a balanced bipartite graph of order $2n$, let $S$ be a subset of $X$ with $|S|\geq 2k+2$, and let $S_{C_i}$ denote the vertices of $S$ on a cycle $C_i$. Prescribed…

Let $G[X,Y]$ be a balanced bipartite graph of order $2n$, let $S$ be a subset of $X$ with $|S|\geq 2k+1$, and let $\sigma_{1,1}(S)$ denote the degree-sum parameter used in the pape…

Weakly supercritical circumference conjecture.

Linear circumference conjecture.

Let $H$ and $H'$ be graphs with $m_2(H)\geq m_2(H')$. Asymmetric Ramsey-density conjecture. If neither $H$ nor $H'$ is a matching and at least one of $H$ or $H'$ contains a cycle,…

Let $G=(V,E)$ be a graph, and let a $2$-cycle on $G$ be a bilinear form in the relevant $2$-cycle module $L(G)$. Circuit-pair $2$-cycles are those arising from vertex-disjoint orie…

Let $k>1$, and let $C_{2k+1}$ denote the cycle of length $2k+1$. A graph is $C_{2k+1}$-free if it contains no subgraph isomorphic to $C_{2k+1}$; $K_{a,b}$ denotes the complete bipa…

Let $G$ be a graph, let $\overline{G}$ denote its complement, and for $k\geq 3$ let $f_k(G)$ be the number of frustrated $k$-cycles, where a cyclic ordering $v_1v_2\dots v_k$ is fr…

Let $T(n,2)$ denote the graph used in the paper's notation for the relevant two-part construction, and let $T(n,2)$ be cycle-maximal when it is triangle-free and no added edge pres…

Let a cycle-maximal triangle-free graph be a triangle-free graph to which no edge can be added without creating a triangle. For a graph on $n$ vertices, let…

Let $c$ be the minimum constant such that every digraph of order $n$ with minimum out-degree at least $cn$ contains a directed cycle of length at most $3$. Let $c'$ be the minimum…