A simple and elementary proof of Whitney's unique embedding theorem

Whitney's Theorem for Line Graphs of Multi-Graphs

Whitney's Theorem states that every graph, different from $K_3$ or $K_{1,3}$, is uniquely determined by its line graph. A $1$-line graph of a multi-graph is the graph with as vertices the edges of the multi-graph, and two edges adjacent if and only if there is a unique vertex on both edges. The $\geq 1$-line graph of a multi-graph is the graph on the edges of the multi-graph, where two edges are adjacent if and only if there is at least one vertex on both edges. We extend Whitney's theorem to such line graphs of multi-graphs, and show that most multi-graphs are uniquely determined by their lin…

Some Theorems and Applications of the $(q,r)$-Whitney Numbers

The $(q,r)$-Whitney numbers were recently defined in terms of the $q$-Boson operators, and several combinatorial properties which appear to be $q$-analogues of similar properties were studied. In this paper, we obtain elementary and complete symmetric polynomial forms for the $(q,r)$-Whitney numbers, and give combinatorial interpretations in the context of $A$-tableaux. We also obtain convolution-type identities using the combinatorics of $A$-tableaux. Lastly, we present applications and theorems related to discrete $q$-distributions.

On graphs uniquely defined by their $k$-circular matroids

In 30's Hassler Whitney considered and completely solved the problem $(WP)$ of describing the classes of graphs $G$ having the same cycle matroid $M(G)$. A natural analog $(WP)'$ of Whitney's problem $(WP)$ is to describe the classes of graphs $G$ having the same matroid $M'(G)$, where $M'(G)$ is a matroid on the edge set of $G$ distinct from $M(G)$. For example, the corresponding problem $(WP)' = (WP)_{θ}$ for the so-called bicircular matroid $M_{θ}(G)$ of graph $G$ was solved by Coulard, Del Greco and Wagner. In our previous paper [arXive:1508.05364] we introduced and studied the so-called $…

$2$-complexes with unique embeddings in 3-space

A well-known theorem of Whitney states that a 3-connected planar graph admits an essentially unique embedding into the 2-sphere. We prove a 3-dimensional analogue: a simply-connected $2$-complex every link graph of which is 3-connected admits an essentially unique locally flat embedding into the 3-sphere, if it admits one at all. This can be thought of as a generalisation of the 3-dimensional Schoenflies theorem.

图中高连通图的matroid刚性和重构

Rigidity and reconstruction in matroids of highly connected graphs

本文研究了图matroid的刚性和重构性质，证明了在无界情况下，Whitney性质与Lovász-Yemini性质等价，并展示了1-可扩展图matroid家族的重构特性。

K-Circular Matroids of Graphs

In 30's Hassler Whitney considered and completely solved the problem $(WP)$ of describing the classes of graphs $G$ having the same cycle matroid $M(G)$. A natural analog $(WP)'$ of Whitney's problem $(WP)$ is to describe the classes of graphs $G$ having the same matroid $M'(G)$, where $M'(G)$ is a matroid (on the edge set of $G$) distinct from $M(G)$. For example, the corresponding problem $(WP)'= (WP)_{θ}$ for the so-called bicircular matroid $M_{θ}(G)$ of graph $G$ was solved by Coulard, Del Greco and Wagner. We define the so-called {\em $k$-circular matroid} $M_k(G)$ on the edge set of gra…