Related papers: Increasing spanning forests in graphs and simplici…

Given a spanning tree $T$ of a planar graph $G$, the co-tree of $T$ is the spanning tree of the dual graph $G^*$ with edge set $(E(G)-E(T))^*$. Gr\"unbaum conjectured in 1970 that every planar 3-connected graph $G$ contains a spanning tree…

An edge-ordered graph is a graph with a total ordering of its edges. A path $P=v_1v_2\ldots v_k$ in an edge-ordered graph is called increasing if $(v_iv_{i+1}) > (v_{i+1}v_{i+2})$ for all $i = 1,\ldots,k-2$; it is called decreasing if…

Given a graph $G=(V,E)$ and a set $C$ of unordered pairs of edges regarded as being in conflict, a stable spanning tree in $G$ is a set of edges $T$ inducing a spanning tree in $G$, such that for each $\left\lbrace e_i, e_j \right\rbrace…

Let $R$ and $B$ be two disjoint sets of points in the plane where the points of $R$ are colored red and the points of $B$ are colored blue, and let $n=|R\cup B|$. A bichromatic spanning tree is a spanning tree in the complete bipartite…

For a poset whose Hasse diagram is a rooted plane forest $F$, we consider the corresponding tree descent polynomial $A_F(q)$, which is a generating function of the number of descents of the labelings of $F$. When the forest is a path,…

In this work, we study the color discrepancy of spanning trees in random graphs. We show that for the Erd\H{o}s-R\'enyi random graph $G(n,p)$ with $p$ above the connectivity threshold, the following holds with high probability: in every…

For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. In this paper, we study the problem of bounding t(G) for graphs which do not contain a complete graph K_r on r vertices. This problem…

Given a class of graphs F, we say that a graph G is universal for F, or F-universal, if every H in F is contained in G as a subgraph. The construction of sparse universal graphs for various families F has received a considerable amount of…

We provide new evidence that spanning forests of graphs satisfy the same negative correlation properties as spanning trees, derived from Lord Rayleigh's monotonicity property for electrical networks. The main result of this paper is that…

Consider the following process on a simple graph without isolated vertices: Order the edges randomly and keep an edge if and only if it contains a vertex which is not contained in some preceding edge. The resulting set of edges forms a…

We prove that the infinite components of the Free Uniform Spanning Forest of a Cayley graph are indistinguishable by any invariant property, given that the forest is different from its wired counterpart. Similar result is obtained for the…

In his Ph.D. thesis, Farber proved that every strongly chordal graph can be represented as intersection graph of subtrees of a weighted tree, and these subtrees are ``compatible''. Moreover, this is an equivalent characterization of…

After the discovery of Khovanov homology, which categorifies the Jones polynomial, an analogous categorification of the chromatic polynomial, known as chromatic homology, was introduced. Its graded Euler characteristic recovers the…

A well-known result due to Caro (1979) and Wei (1981) states that every graph $G$ has an independent set of size at least $\sum_{v\in V(G)} \frac{1}{d(v) + 1}$, where $d(v)$ denotes the degree of vertex $v$. Alon, Kahn, and Seymour (1987)…

Given a multigraph $G$ and function $f : V(G) \rightarrow \mathbb{Z}_{\ge 2}$ on its vertices, a degree-$f$ subgraph of $G$ is a spanning subgraph in which every vertex $v$ has degree at most $f(v)$. The degree-$f$ arboricity $a_f(G)$ of…

In the Properly Colored Spanning Tree problem, we are given an edge-colored undirected graph and the goal is to find a spanning tree in which any two adjacent edges have distinct colors. Since finding such a tree is NP-hard in general,…

Let $G$ be a connected graph and let $k$ be a positive integer. Let $T$ be a spanning tree of $G$. The leaf degree of a vertex $v\in V(T)$ is defined as the number of leaves adjacent to $v$ in $T$. The leaf degree of $T$ is the maximum leaf…

It is well-known that the graphs not containing a given graph H as a subgraph have bounded chromatic number if and only if H is acyclic. Here we consider ordered graphs, i.e., graphs with a linear ordering on their vertex set, and the…

In this paper, we study collective additive tree spanners for families of graphs enjoying special Robertson-Seymour's tree-decompositions, and demonstrate interesting consequences of obtained results. We say that a graph $G$ {\em admits a…

Schmidt characterised the class of rayless graphs by an ordinal rank function, which makes it possible to prove statements about rayless graphs by transfinite induction. Halin asked whether Schmidt's rank function can be generalised to…