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Shuyan Chen

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Preprint Aug 2026

Cyclic Sources of Strong Domination in Graph Norms

Conlon and Lee asked for strongly dominating graphs beyond norming graphs and even paths. We construct a two-parameter family of pairwise non-isomorphic $2$-connected strongly dominating graphs that are not seminorming, and hence lie outside the two classes of examples previously identified for signed strong domination. The construction uses cyclic amalgamation of two-rooted blocks. For root-reversible blocks, we characterize the generation of all even cyclic amalgams by local even-Schatten inequalities for transfer operators. We determine this criterion for $K_{2,m}$, with the roots in the part of size $m$: it holds exactly when $m$ is even. We also classify the connected outerplanar strongly dominating graphs and the connected root-reversible outerplanar blocks satisfying the universal cyclic criterion.

Shuyan Chen · 0 citations

Preprint Aug 2026

Odd-Cycle Span Defect: A Polynomial Lower Bound and a Square-Root Upper Bound

For a graph $G$, let $\psi(G)=\max\{\chi(G[V(C)]):C$ is an odd cycle of $G\}$, with $\psi(G)=0$ when $G$ is bipartite. For positive integers $N$, set $F(N)=\max\{\chi(G)-\psi(G):|V(G)|\le N\}$. The function $F$ measures the finite-order additive gap arising from an open problem of Erdos and Hajnal. We prove $N^{1/6-o(1)}\le F(N)<\sqrt{6N}$. The lower bound raises the finite-order scale supplied by the Cameron-Clow path-colour construction from $\log N/\log\log N$ to a fixed power of $N$. Its proof constructs a palette-code graph from a binary covering code $\mathcal{C}\subseteq\{0,1\}^p$ and establishes the exact identities $\chi(G)=2p+\ell-\rho(\mathcal{C})$ and $\psi(G)=2p$. Near-middle Hamming coverings yield the exponent $1/6$. The upper bound combines Polavarapu's connectivity theorem, the Chvatal-Erdos Hamiltonicity theorem, and maximum-independent-set stripping.

Shuyan Chen · 0 citations