Related papers: Partial-duals for planar ribbon graphs

We prove that the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the dimer partition…

We introduce an additional structure on ribbon graphs, arrow structure. We extend the Bollob\'as-Riordan polynomial to ribbon graph with this structure. The extended polynomial satisfies the contraction-deletion relations and naturally…

Let $G$ be a graph with adjacency matrix $A(G)$ and degree matrix $D(G)$, and let $L_\mu(G):=A(G)-\mu D(G)$. Two graphs $G_1$ and $G_2$ are called \emph{degree-similar} if there exists an invertible matrix $M$ such that $M^{-1} A(G_1) M…

We study the existence of perfect matchings in suitably chosen induced subgraphs of random biregular bipartite graphs. We prove a result similar to a classical theorem of Erdos and Renyi about perfect matchings in random bipartite graphs.…

For any graph G = (V, E) and proportion $p\in(0,1]$, a set $S\subseteq V$ is a p-dominating set if $\frac{|N[S]|}{|V|}\geq p$. The $p$-domination number $\gamma_{p}(G)$ equals the minimum cardinality of a $p$-dominating set in G. For a…

Given a graph $G$ and $p \in [0,1]$, let $G_p$ denote the random subgraph of $G$ obtained by keeping each edge independently with probability $p$. Alon, Krivelevich, and Sudokov proved $\mathbb{E} [\chi(G_p)] \geq C_p \frac{\chi(G)}{\log…

In this paper, two recursion formulae of chromatic polynomial of a maximal planar graph G are obtained. Moreover, the application of these formulaes to the proof of Four-Color Conjecture is investigated. By using these formulae, the proof…

We prove two recent conjectures on some upper bounds for the Castelnuovo-Mumford regularity of the binomial edge ideals of some different classes of graphs. We prove the conjecture of Matsuda and Murai for graphs which has a cut edge or a…

We define a knot to be half ribbon if it is the cross-section of a ribbon 2-knot, and observe that ribbon implies half ribbon implies slice. We introduce the half ribbon genus of a knot K, the minimum genus of a ribbon knotted surface of…

Inspired by a width invariant on permutations defined by Guillemot and Marx, Bonnet, Kim, Thomass\'e, and Watrigant introduced the twin-width of graphs, which is a parameter describing its structural complexity. This invariant has been…

In this purely experimental work we try to represent the set of plane maps with 3 vertices and 3 faces as a bipartite ribbon graph. In particular, this construction allows one to estimate the genus of the initial set.

We study the problem of finding the minimal (maximal) genus for a surface where a given four-valent graph with fixed opposite edge structure can be embedded into. We find several partial relations and give new reformulations in…

Let $B=(X,Y,E)$ be a bipartite graph. A half-square of $B$ has one color class of $B$ as vertex set, say $X$; two vertices are adjacent whenever they have a common neighbor in $Y$. Every planar graph is a half-square of a planar bipartite…

In Graph Minor III, Robertson and Seymour conjecture that the tree-width of a planar graph and that of its dual differ by at most one. We prove that given a hypergraph H on a surface of Euler genus k, the tree-width of H^* is at most the…

Here we prove that counting maximum matchings in planar, bipartite graphs is #P-complete. This is somewhat surprising in the light that the number of perfect matchings in planar graphs can be computed in polynomial time. We also prove that…

In this paper we estimate the planar Tur\'an number $\mathrm{ex}_\mathcal{P}(n,H)$ of some graphs $H$, i.e., the maximum number of edges in a planar graph $G$ of $n$ vertices not containing $H$ as a subgraph. We give a new, short proof when…

In "Bipartite minors" [Journal of Combinatorial Theory, Series B, 2016], Chudnovsky et al. introduced the bipartite minor relation, a quasi-order on the class of bipartite graphs somewhat analogous the minor relation on general graphs and…

A bipartite graph $G=(V,E)$ with $V=V_1\cup V_2$ is biregular if all the vertices of a stable set $V_i$ have the same degree $r_i$ for $i=1,2$. In this paper, we give an improved new Moore bound for an infinite family of such graphs with…

We present those properties of planar doodles, especially when regarded as 4-valent graphs, that enable us to classify them into {\it prime} and {\it super prime} doodles by analogy to a knot sum. We describe a method for partially…

We propose a description of T-duality between general geometric and non-geometric backgrounds as higher groupoid bundles with connections. Our description extends the previous observation by Nikolaus and Waldorf that the topological aspects…