13 problems

Order-bound conjecture. Then

Generalized spectral closedness conjecture for split graphs. The class $\operatorname{Forb}(\mathcal{F})$ is generalized spectrally closed, but it admits no walk-realizable…

Linear-time certifying algorithm conjecture. For every fixed positive integer $k$, there is a certifying algorithm that runs in time

Let $p$ and $q$ be distinct primes, and let $x,y\in\mathbb{N}$. Consider natural numbers of the form … A natural number is $\Delta$-primitive if it has the $\Delta$ property and is…

A natural number is called $\Delta$-primitive when it has the $\Delta$ property but is not a nontrivial square multiple of another number with the $\Delta$ property. Infinitude con…

Let $s\in\mathbb{N}$ be square-free, and let $P(s)$ be the set of $\Delta$-primitives with square-free part $s$. A natural number $n$ has a $\Delta$-primitive decomposition if it c…

Dai–Zhang–Broerama–Zhang conjecture. Every $(r-1)$-connected $K_{1,r}$-free split graph is Hamiltonian.

Let $G$ be a co-chordal graph. The Lyu–Hicks conjecture. The biclique partition number satisfies … where $\operatorname{mc}(G^c)$ denotes the number of maximal cliques in the compl…

Generalized Hamilton-connectedness conjecture. Every $r$-connected ${K_{1,r+1},K_{1,r+1}+e}$-free split graph of order at least $2r+7$ is Hamilton-connected.

Hilton's conjecture. If $\Delta(G)>\frac{|V|}{3}$, then $G$ is Class 1 if and only if it is not subgraph-overfull. The supplied paper states that this conjecture holds for the clas…

Let $R_k^{\mathcal{SP}}(i,j)$ denote the $k$-defective Ramsey number for split graphs. Split-graph defective Ramsey conjecture. For all $i,j\geq k+2$, … The formula is proposed to…

Let $u_n$ be the number of unbalanced split graphs on $n$ vertices, and let $s_n$ be the number of split graphs on $n$ vertices. The ratio of balanced split graphs to split graphs…

Let $G$ be a free-twin split graph, meaning that $G$ is a split graph with no pair of free twins. Write $\chi_{rlid}(G)$ for its relaxed locally identifying chromatic number and…