Related papers: Spanning Triangle-trees and Flows of Graphs

We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have been recently applied to obtain various properties of…

In 2001, Koml\'os, S\'ark\"ozy, and Szemer\'edi proved that every sufficiently large $n$-vertex graph with minimum degree at least $\left(1/2+\gamma\right)n$ contains all spanning trees with maximum degree at most $cn/\log n$. We extend…

We investigate flows on graphs whose links have random capacities. For binary trees we derive the probability distribution for the maximal flow from the root to a leaf, and show that for infinite trees it vanishes beyond a certain threshold…

The celebrated Nash-Williams and Tutte's theorem states that a graph $G=(V, E)$ contains $k$ edge disjoint spanning trees if and only if $\nu_{f}(G) \geq k$, where $$\nu_{f}(G):=\min_{|\mathcal{\mathcal{P}}|>1, \text{$\mathcal{P}$ is a…

Let $R$ and $B$ be a set of red points and a set of blue points in the plane, respectively, such that $R\cup B$ is in general position, and let $f:R \to \{2,3,4, \ldots \}$ be a function. We show that if $2\le |B|\le \sum_{x\in R}(f(x)-2) +…

We consider infinite connected quasi-transitive locally finite graphs and show that every such graph with more than one end is a tree amalgamation of two other such graphs. This can be seen as a graph-theoretical version of Stallings'…

A conjecture of Berge suggests that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. Since three perfect matchings suffice only when the graph in question is $3$-edge-colourable, the rest of cubic…

We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected…

Let $T$ be a tree, a vertex of degree one is called a leaf. The set of leaves of $T$ is denoted by $Leaf(T)$. The subtree $T-Leaf(T)$ of $T$ is called the stem of $T$ and denoted by $Stem(T).$ In this note, we give a sharp sufficient…

Let $G$ be a bridgeless graph, $C$ is a circuit of $G$. Fan proposed a conjecture that if $G/C$ admits a nowhere-zero 4-flow, then $G$ admits a 4-flow $(D,f)$ such that $E(G)-E(C)\subseteq$ supp$(f)$ and $|\textrm{supp}(f)\cap…

In a series of three papers we develop an end space theory for digraphs. Here in the third paper we introduce a concept of depth-first search trees in infinite digraphs, which we call normal spanning arborescences. We show that normal…

For any integer $k\geq1,$ a graph $G$ has a $k$-factor if it contains a $k$-regular spanning subgraph. In this paper we prove a sufficient condition in terms of the number of $r$-cliques to guarantee the existence of a $k$-factor in a graph…

The Erd\H{o}s-S\'os Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with $k$ edges as a subgraph. We prove that there are $\delta>0$ and $k_0\in\mathbb N$ such that the conjecture holds for every…

The triangle-free process begins with an empty graph on n vertices and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. We determine the asymptotic number of edges in the maximal…

Flows and colorings are disparate concepts in graph algorithms -- the former is tractable while the latter is intractable. Tutte introduced the concept of nowhere-zero flows to unify these two concepts. Jaeger showed that nowhere-zero flows…

Motivated by a result of [1] which states that if F is a subgraph of a convex complete graph K_n and F contains no boundary edge of K_n and |E(F)| \leq n-3, then K_n - F admits a triangulation, we determine necessary and sufficient…

We consider the problem of how much edge connectivity is necessary to force a graph G to contain a fixed graph H as an immersion. We show that if the maximum degree in H is D, then all the examples of D-edge connected graphs which do not…

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree…

Any $n$-vertex $3$-graph with minimum codegree at least $\lfloor n/3\rfloor$ must have a spanning tight component, but immediately below this threshold it is possible for no tight component to span more than $\lceil 2n/3\rceil$ vertices.…

Let $G$ be a bridgeless cubic graph, and $\mu_2(G)$ the minimum number $k$ such that two 1-factors of $G$ intersect in $k$ edges. A cyclically $n$-edge-connected cubic graph $G$ has a nowhere-zero 5-flow if (1) $n \geq 6$ and $\mu_2(G) \leq…