44 problems

Let $G$ be a chordal graph, and let $J(G)$ be its cover ideal. A graded ideal is componentwise linear when each ideal generated by its elements of a fixed degree has a linear resol…

Hendry's conjecture. Every Hamiltonian chordal graph is fully cycle extendible.

Let $G$ be a chordal graph, meaning that every cycle of length at least four has a chord. For an integer $k\geq1$, let $G^k$ be the $k$th graph power, in which two vertices are adj…

Loebl's conjectures. The q-dichromate is equivalent to the U-polynomial, and the q-dichromate distinguishes non-isomorphic chordal graphs. Equivalently, in the latter assertion, eq…

Let $G$ be a chordal graph, and let $\operatorname{dmg}(G)$ and $\operatorname{zdmg}(G)$ denote its ordinary and zombie damage numbers. Let $\gamma_c(G)$ be its connected dominatio…

Let $G$ be a connected chordal graph, meaning that it has no induced cycle of length greater than $3$, and let $\rho\ge 2$. A weak retract of $G$ is a graph obtained from $G$ by a…

Dallard et al.'s conjecture. There exists no minimally $t$-tough chordal graph for any real number $t>1/2$.

Let $G$ be a finite graph, and call it $r$-locally chordal if every ball of radius $r$ in $G$ is chordal. Let $|V(G)|$ denote the number of vertices of $G$ and let…

Bounded-mad backbone colouring conjecture.

Backbone colouring conjecture.

Let $G$ be a chordal graph, let $I(G)$ be its edge ideal, let $I(G)^{[k]}$ denote its $k$-th square-free power, let $\nu(G)$ be its matching number, and let $\mathrm{aim}(G,k)$ be…

Simplicial-support conjecture. There is a $D$-unsolvable configuration $C^*$ such that

Let $D=(V,A)$ be a digraph whose underlying undirected graph is $5$-chordal, meaning that it has no chordless cycle of length more than $5$. A dicut is a set of arcs directed acros…

Let $G$ be a connected chordal graph admitting a tree representation $T$, where $T$ is a subdivided caterpillar. Subdivided-caterpillar Gallai conjecture. Then … This asserts that…

Let $G$ be an undirected graph, and let $P_G$ denote the spanning tree generating function corresponding to $G$. A polynomial is homaloidal if its polar map is birational. Chordali…

For an integer $k\geq 1$, a $k$-forest is a $K_{k+2}$-free chordal graph; in particular, a $2$-forest is a $K_4$-free chordal graph. An even-hole-free graph has no induced cycle of…

Let $t\geq 4$ be an integer, let $H$ be a graph, and call $H$ $K_4$-free chordal if it is chordal and contains no induced subgraph isomorphic to $K_4$. An even-hole-free graph has…

Let $G$ be a graph. It is even-hole-free if it has no induced cycle of even length, and it is chordal if it has no induced cycle of length at least four. Sivaraman's conjecture. If…

Let $P$ be a polytope with vertex set $V$. Two vertices $alpha,beta\in V$ fulfill the rhombus criterion if either $\operatorname{conv}(\alpha,\beta)$ is an edge of $P$, or there ex…

Figueiredo et al.'s conjecture. $G$ is Class 2 if and only if it is subgraph-overfull. This conjecture proposes a characterization of edge-chromatic Class 2 chordal graphs by the s…

Reduced clique graph recognition conjecture. There is a polynomial-time algorithm for deciding whether a given graph is isomorphic to $C_R(G)$ for some chordal graph $G$.

Chordal graph representation conjecture. Let $H$ be a chordal graph. There are chordal graphs $G$ and $G'$ such that $H$ is isomorphic to both $C(G)$ and $C_R(G')$.

Seven-hole conjecture. There is no chordal graph $G$ such that $C_R(G)$ contains an induced cycle with seven or more vertices.

Induced-cycle bound conjecture. A reduced clique graph cannot have an induced cycle of length greater than six; consequently, a rotunda graph cannot have an induced cycle of length…

No minimally tough chordal graph conjecture. There exists no minimally $t$-tough chordal graph.