Related papers: Elementary proof for the bounds of the complexity …
The number of spanning trees of a graph $G$, denoted $\tau(G)$, is a well studied graph parameter with numerous connections to other areas of mathematics. In a recent remarkable paper, answering a question of Sedl\'a\v{c}ek from 1969, Chan,…
We give a proof for sharp estimate for the number of spanning trees using linear algebra and generalize this bound to multigraphs. In addition, we show that this bound is tight for complete graphs. In addition, we give estimates for number…
We prove that every connected graph with $s$ vertices of degree~1 and 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${1\over 3}t +{1\over 4}s+{3\over 2}$ leaves. We present infinite series of graphs showing that…
We prove that any planar graph on $n$ vertices has less than $O(5{.}2852^n)$ spanning trees. Under the restriction that the planar graph is 3-connected and contains no triangle and no quadrilateral the number of its spanning trees is less…
We prove, that every connected graph with $s$ vertices of degree 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${2\over 5}t +{1\over 5}s+\alpha$ leaves, where $\alpha \ge {8\over 5}$. Moreover, $\alpha \ge 2$ for…
We define the \emph{visual complexity} of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to…
Let $\mathcal{G}$ be the set of simple graphs (or multigraphs) $G$ such that for each $G \in \mathcal{G}$ there exists at least two non-empty disjoint proper subsets $V_{1},V_{2}\subseteq V(G)$ satisfying $V(G)\setminus(V_{1} \cup…
Tuza conjectured that for every graph $G$, the maximum size $\nu$ of a set of edge-disjoint triangles and minimum size $\tau$ of a set of edges meeting all triangles satisfy $\tau \leq 2\nu$. We consider an edge-weighted version of this…
Let $G_1$ and $G_2$ be simple graphs and let $n_1 = |V(G_1)|$, $m_1 = |E(G_1)|$, $n_2 = |V(G_2)|$ and $m_2 = |E(G_2)|.$ In this paper we derive sharp upper and lower bounds for the number of spanning trees $\tau$ in the Cartesian product…
The visual complexity of a graph drawing can be measured by the number of geometric objects used for the representation of its elements. In this paper, we study planar graph drawings where edges are represented by few segments. In such a…
For a graph $G$, let $\tau(G)$ denote the number of spanning trees. We show that for every fixed $0 < c < 1/4$, the number of distinct values of $\tau(G)$, as $G$ ranges over simple graphs on $n$ vertices, is at least $\exp(c n \log n)$ for…
In this paper we give an exact analytical expression for the number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs. This number is an important graph invariant related to different topological…
It is proved that the rectilinear crossing number of every graph with bounded tree-width and bounded degree is linear in the number of vertices. **** This paper has been withdrawn by the author. **** The results have been superseeded by the…
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…
Tuza (1981) conjectured that the size $\tau(G)$ of a minimum set of edges that intersects every triangle of a graph $G$ is at most twice the size $\nu(G)$ of a maximum set of edge-disjoint triangles of $G$. In this paper we present three…
For any connected multigraph $G=(V,E)$ and any $M\subseteq E$, if $M$ induces an acyclic subgraph of $G$ and removing all edges in $M$ yields a subgraph of $G$ whose components are complete graphs, a formula for $\tau_G(M)$ is obtained,…
We prove that every connected graph with $s$ vertices of degree not 2 has a spanning tree with at least ${1\over 4}(s-2)+2$ leaves. Let $G$ be a be a connected graph of girth $g$ with $v>1$ vertices. Let maximal chain of successively…
In Graph Minors III, Robertson and Seymour write: "It seems that the tree-width of a planar graph and the tree-width of its geometric dual are approximately equal - indeed, we have convinced ourselves that they differ by at most one". They…
Among subgraphs with a fixed number of vertices of the regular square lattice, we prove inequalities that essentially say that those with smaller boundaries have larger numbers of spanning trees and vice-versa. As an application, we relate…
The complexity of a finite connected graph is its number of spanning trees; for a non-connected graph it is the product of complexities of its connected components. If $G$ is an infinite graph with cofinite free ${\mathbb Z}^d$-symmetry,…