Author

Ilkyoo Choi

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Preprint Aug 2026

Induced-saturated graphs exist for even cycles

A graph $G$ is \emph{$H$-induced-saturated} if $G$ has no induced subgraph isomorphic to $H$ but changing the adjacency of an arbitrary pair of vertices in $G$ creates an induced copy of $H$. The existence problem for $H$-induced-saturated graphs had previously been settled when $H$ is a complete graph, a path, an odd cycle, or an even cycle of length at most $10$. In this paper, for every integer $q\ge3$, we construct a $C_{2q+2}$-induced-saturated graph. Hence, induced-saturated graphs exist for all cycles, except for the cycle of length 3.

Ilkyoo Choi · 0 citations

Open access Jul 2026

On Sparsity Conditions Guaranteeing a Fractional Coloring

A graph has an ‐coloring if there exists an assignment from the vertices to subsets of with size such that adjacent vertices are assigned disjoint subsets. Odd girth at least is a necessary condition for a graph to have a ‐coloring. Chen and Raspaud conjectured a tight upper bound on the maximum average degree of a graph with odd girth at least that guarantees a ‐coloring. Namely, they conjectured that every graph with odd girth at least and maximum average degree less than has a ‐coloring. This conjecture is true for ; when , computers were used to perform case analysis. The main result of this paper confirms the conjecture for the next open case () without the use of computers. Moreover, our approach yields simpler, computer‐free proofs for previously known cases ().

Ilkyoo Choi · Journal of Graph Theory · 1 citation · ⚡1