Related papers: A De Bruijn-Erdos theorem for chordal graphs
The regular embeddings of complete bipartite graphs $K_{n,n}$ in orientable surfaces are classified and enumerated, and their automorphism groups and combinatorial properties are determined. The method depends on earlier classifications in…
Some well-known results of Prodinger and Tichy are that the number of independent sets in the $n$-vertex path graph is $F_{n+2}$, and that the number of independent sets in the $n$-vertex cycle graph is $L_n$. We generalize these results by…
Let $L$ be a sequence $(\ell_1,\ell_2,\ldots,\ell_n)$ of $n$ lines in $\mathbb{C}^3$. We define the {\it intersection graph} $G_L=([n],E)$ of $L$, where $[n]:=\{1,\ldots, n\}$, and with $\{i,j\}\in E$ if and only if $i\neq j$ and the…
Plural (or multiple-conclusion) cuts are inferences made by applying a structural rule introduced by Gentzen for his sequent formulation of classical logic. As singular (single-conclusion) cuts yield trees, which underlie ordinary natural…
We discuss the problem of embedding graphs in the plane with restrictions on the vertex mapping. In particular, we introduce a technique for drawing planar graphs with a fixed vertex mapping that bounds the number of times edges bend. An…
This paper has been withdrawn by the author. Peterson and Woodall previously proved that the list-edge-colouring conjecture holds for graphs without odd cycles of length 5 or longer. D. Peterson and D. R. Woodall, Edge-choosability in…
Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and Kazakov and Korablev proved that for every spatial complete graph…
A property of finite graphs is called nondeterministically testable if it has a "certificate" such that once the certificate is specified, its correctness can be verified by random local testing. In this paper we study certificates that…
The interplay between algebro-geometric and combinatorial Brill-Noether theory is studied. The Brill-Noether variety of a graph shown to be non-empty if the Brill-Noether number is non-negative, as a consequence of the analogous fact for…
We prove the invariance of scalar Feynman graphs of any planar topology under the Yangian level-one momentum symmetry given certain constraints on the propagator powers. The proof relies on relating this symmetry to a planarized version of…
In the random geometric graph $G(n,r_n)$, $n$ vertices are placed randomly in Euclidean $d$-space and edges are added between any pair of vertices distant at most $r_n$ from each other. We establish strong laws of large numbers (LLNs) for a…
A recent framework for generalizing the Erdos-Ko-Rado Theorem, due to Holroyd, Spencer, and Talbot, defines the Erdos-Ko-Rado property for a graph in terms of the graph's independent sets. Since the family of all independent sets of a graph…
The divisor theory of the complete graph $K_n$ is in many ways similar to that of a plane curve of degree $n$. We compute the splitting types of all divisors on the complete graph $K_n$. We see that the possible splitting types of divisors…
We prove that for every k, there exists $c_k>0$ such that every graph G on n vertices not inducing a path $P_k$ and its complement contains a clique or a stable set of size $n^{c_k}$.
Let the join of two graphs be the union of two disjoint graphs connected by $j$ edges in a one-to-one manner. In previous work by Gyurov and Pinzon, which generalized the results of Badura and Rara, the determinant of the adjacency matrix…
We prove that for every integer $k$, there exists $\varepsilon > 0$ such that for every n-vertex graph $G$ with no pivot-minor isomorphic to $C_k$, there exist disjoint sets $A,B \subseteq V(G)$ such that $|A|,|B| \geq \varepsilon n$, and…
Chord diagrams and related enlacement graphs of alternating knots are enhanced to obtain complete invariant graphs including chirality detection. Moreover, the equivalence by common enlacement graph is specified and the neighborhood graph…
Motivated by a well-known result in extremal set theory, due to Nicolaas Govert de Bruijn and Paul Erd\H{o}s, we consider curves in the unit $n$-cube $[0,1]^n$ of the form \[ A=\{(x,f_1(x),\ldots,f_{n-2}(x),\alpha): x\in [0,1]\}, \] where…
We address several related problems on combinatorial discrepancy of trees in a setting introduced by Erd\H{o}s, F\"{u}redi, Loebl and S\'{o}s. Given a fixed tree $T$ on $n$ vertices and an edge-colouring of the complete graph $K_n$, for…
We prove a general existence theorem for nonlinear partial differential systems of any order in one complex variable. A special case of first order contains a well-known theorem of Nijenhuis and Woolf concerning local existence of…