arXiv subjects

Qiming Fang

Publications and source records attributed to Qiming Fang.

3 recordsLinked to original sources

arXiv2026

Eulerian-spanning set and coboundary operator: An investigation of maxcut beyond planar graphs

Using the concepts of Eulerian-spanning set and coboundary operator, we generalize Hadlock's conversion of the maxcut problem on planar graphs to one on general graphs with non-negative weights. Using our conversion, we can explore algorithms for maxcut beyond the class of planar graphs. We obtain a Fixed-Parameter Tractable algorithm for $k$-contraction apex graphs. Specifically, our algorithm can be applied to graphs with crossing number $k$, giving an $O(2^k(n+k)^{3/2}\log (n+k))$-time algorithm that matches the best known results when restricted to non-negative weights.

cs.DS↗

arXiv2025

A High-Dimensional Extension of Wagner's Theorem and the Geometrization of Hypergraphs

This paper introduces a geometric representation of hypergraphs by representing hyperedges as simplices. Building on this framework, we employ homotopy groups to analyze the topological structure of hypergraphs embedded in high-dimensional Euclidean spaces. Under the assumptions of the triangulation and that all $i$-th homotopy groups are trivial for $i \leq d-2$, we provide a necessary and sufficient condition for a $d$-uniform hypergraph to be embeddable in $\mathbb{R}^d$, which can be regarded as a kind of high-dimensional extension of Wagner's Theorem for planar graphs. Specifically, we establish that a triangulated $d$-uniform topological hypergraph embeds into $\mathbb{R}^d$ if and only if it contains neither $K_{d+3}^d$ nor $K_{3,d+1}^d$ as a minor. Here, a triangulated $d$-uniform topological hypergraph constitutes a geometrized form of a $d$-uniform hypergraph, while $K_{d+3}^d$ and $K_{3,d+1}^d$ are the high-dimensional generalizations of the complete graph $K_5$ and the complete bipartite graph $K_{3,3}$ in $\mathbb{R}^d$, respectively.

math.CO↗

arXiv2024

Geometrization of Graphs: Towards Bounding the Chromatic Number via High-Dimensional Embedding

We establish a geometric framework by transforming a graph $G$ into a $(d-1)$-dimensional CW complex $U^{d-1}(G)$. This construction is achieved by systematically attaching $i$-spheres ($2 \le i \le d-1$) to $G$ according to specific rules, ensuring that the $j$-th homotopy group of $U^{d-1}(G)$ are trivial for $j = 0, 1, \dots, d-2$. Building upon this construction, we provide a necessary and sufficient condition for $U^{d-1}(G)$ to be embeddable into $\mathbb{R}^d$, which yields an upper bound for the chromatic number $\chi(G)$. To be more specific, we prove that if $G$ does not contain $K_{d+3}$ and $K_{i, d+4-i}$ ($i \in \{2, 3, \dots, \lfloor \frac{d+4}{2} \rfloor \}$) as a minor, then $U^{d-1}(G)$ embeds into $\mathbb{R}^d$ and $\chi(G) \leq 3\cdot 2^{d-1}$. Finally, as a preliminary attempt, we extend the Discharging method to $\mathbb{R}^d$ and investigate the coloring problem for $(d-2)$-faces in $\mathbb{R}^d$.

math.CO↗