Related papers: Counting Circuit Double Covers

We study the topic of "extremal" planar graphs, defining $\mathrm{ex_{_{\mathcal{P}}}}(n,H)$ to be the maximum number of edges possible in a planar graph on $n$ vertices that does not contain a given graph $H$ as a subgraph. In…

A planar graph is essentially $4$-connected if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Jackson and Wormald proved that every essentially 4-connected planar graph $G$ on $n$ vertices contains a…

For large $n$ we determine the maximum number of induced 6-cycles which can be contained in a planar graph on $n$ vertices, and we classify the graphs which achieve this maximum. In particular we show that the maximum is achieved by the…

We revisit the problem of counting the number of copies of a fixed graph in a random graph or multigraph, for various models of random (multi)graphs. For our proofs we introduce the notion of \emph{patchworks} to describe the possible…

A bipartite graph $G = (X \cup Y, E)$ is a 2-layer $k$-planar graph if it admits a drawing on the plane such that the vertices in $X$ and $Y$ are placed on two parallel lines respectively, edges are drawn as straight-line segments, and…

In this paper, for each graph G, a free edge set F is defined. To study the existence of cycle double cover, the naive cycle double cover of G and F have been defined and studied. In the main theorem, the paper, based on the Kuratowski…

We study two parameters obtained from the Euler characteristic by replacing the number of faces with that of induced and induced non-separating cycles. By establishing monotonicity of such parameters under certain homomorphism and edge…

The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all the vertices. It is known that the cover time on any n-vertex, connected graph is at least (1+o(1)) n…

A long-standing Conjecture of S. Negami states that a connected graph has a finite planar cover if and only if it embeds in the projective plane. It is known that the Conjecture is equivalent to the fact that \emph{the graph $K_{1,2, 2, 2}$…

The fascinating question of the maximum value of twin-width on planar graphs is nowadays not far from the final resolution; there is a lower bound of 7 coming from a construction by Kr\'al' and Lamaison [arXiv, September 2022], and an upper…

A grid drawing of a graph maps vertices to grid points and edges to line segments that avoid grid points representing other vertices. We show that there is a number of grid points that some line segment of an arbitrary grid drawing must…

We give a series of new lower bounds on the minimum number of vertices required by a graph to contain every graph of a given family as induced subgraph. In particular, we show that this induced-universal graph for $n$-vertex planar graphs…

An embedding of a graph in $3$-space is linkless if for every two disjoint cycles there exists an embedded ball that contains one of the cycles and is disjoint from the other. We prove that every bipartite linklessly embeddable (simple)…

It is proved that for $n \geq 6$, the number of perfect matchings in a simple connected cubic graph on $2n$ vertices is at most $4 f_{n-1}$, with $f_n$ being the $n$-th Fibonacci number. The unique extremal graph is characterized as well.…

The $2$-layer drawing model is a well-established paradigm to visualize bipartite graphs. Several beyond-planar graph classes have been studied under this model. Surprisingly, however, the fundamental class of $k$-planar graphs has been…

Let $G$ be a plane graph with outer cycle $C$ and let $(L(v):v\in V(G))$ be a family of sets such that $|L(v)|\ge 5$ for every $v\in V(G)$. By an $L$-coloring of a subgraph $J$ of $G$ we mean a (proper) coloring $\phi$ of $J$ such that…

A classical result of Bondy and Simonovits in extremal graph theory states that if a graph on $n$ vertices contains no cycle of length $2k$ then it has at most $O(n^{1+1/k})$ edges. However, matching lower bounds are only known for…

In a graph $G$, a vertex dominates itself and its neighbors. A subset $S$ of vertices of $G$ is a double dominating set of $G$ if every vertex is dominated by at least two vertices in $S$. The double domination number $\gamma_{\times 2}(G)$…

The motivation for this paper is to study the complexity of constant-width arithmetic circuits. Our main results are the following. 1. For every k > 1, we provide an explicit polynomial that can be computed by a linear-sized monotone…

Except for Koshy who devotes seven pages to applications of Fibonacci Numbers to electric circuits, most books and the Fibonacci Quarterly have been relatively silent on applications of graphs and electric circuits to Fibonacci numbers.…