Related papers: Linear arboricity of degenerate graphs

For a set of graphs $\mathcal{F}$, a graph is said to be $\mathcal{F}$-free if it does not contain any graph in $\mathcal{F}$ as a subgraph. Let Ex$_{sp}(n,\mathcal{F})$ denote the graphs with the maximum spectral radius among all…

Let $G=(V,E)$ be a graph on $n$ vertices, and let $\lambda_1(L(G))\ge \cdots\ge \lambda_{n-1}(L(G))\ge \lambda_n(L(G))=0$ be the eigenvalues of its Laplacian matrix $L(G)$. Brouwer conjectured that for every $1\le k\le n$, $\sum_{i=1}^k…

The product dimension of a graph G is defined as the minimum natural number l such that G is an induced subgraph of a direct product of l complete graphs. In this paper we study the product dimension of forests, bounded treewidth graphs and…

Tree-decompositions and treewidth are of fundamental importance in structural and algorithmic graph theory. The "spread" of a tree-decomposition is the minimum integer $s$ such that every vertex lies in at most $s$ bags. A…

A graph is $k$-degenerate if every subgraph $H$ has a vertex $v$ with $d_{H}(v) \leq k$. The class of degenerate graphs plays an important role in the graph coloring theory. Observed that every $k$-degenerate graph is $(k + 1)$-choosable…

We define the induced arboricity of a graph $G$, denoted by ${\rm ia}(G)$, as the smallest $k$ such that the edges of $G$ can be covered with $k$ induced forests in $G$. This notion generalizes the classical notions of the arboricity and…

A path partition (also referred to as a linear forest) of a graph $G$ is a set of vertex-disjoint paths which together contain all the vertices of $G$. An isolated vertex is considered to be a path in this case. The path partition…

For any graph $G$, let $t(G)$ be the number of spanning trees of $G$, $L(G)$ be the line graph of $G$ and for any non-negative integer $r$, $S_r(G)$ be the graph obtained from $G$ by replacing each edge $e$ by a path of length $r+1$…

Linear rank-width is a graph width parameter, which is a variation of rank-width by restricting its tree to a caterpillar. As a corollary of known theorems, for each $k$, there is a finite obstruction set $\mathcal{O}_k$ of graphs such that…

A $k$-box $B=(R_1,...,R_k)$, where each $R_i$ is a closed interval on the real line, is defined to be the Cartesian product $R_1\times R_2\times ...\times R_k$. If each $R_i$ is a unit length interval, we call $B$ a $k$-cube. Boxicity of a…

An acyclic edge coloring of a graph is a proper edge coloring without any bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum $k$ such that $G$ has an acyclic edge coloring with $k$ colors.…

For an integer $k$ at least $2$, and a graph $G$, let $f_k(G)$ be the minimum cardinality of a set $X$ of vertices of $G$ such that $G-X$ has either $k$ vertices of maximum degree or order less than $k$. Caro and Yuster (Discrete…

The Alon-Tarsi number of a graph $ G $ is the smallest $ k $ such that there exists an orientation $ D $ of $ G $ with maximum outdegree $ k - 1 $ satisfying that the number of even Eulerian subgraphs is different from the number of odd…

A tree with at most k leaves is called k-ended tree, and a tree with exactly k leaves is called k-end tree, where a leaf is a vertex of degree one. Contraction of a graph G along the edge e means deleting the edge e and identifying its end…

A vertex of degree one is called an end-vertex, and an end-vertex of a tree is called a leaf. A tree with at most $k$ leaves is called a $k$-ended tree. For a positive integer $k$, let $t_k$ be the order of a largest $k$-ended tree. Let…

Let $G = (V, E)$ be a connected graph with maximum degree $k\geq 3$ distinct from $K_{k+1}$. Given integers $s \geq 2$ and $p_1,\ldots,p_s\geq 0$, $G$ is said to be $(p_1, \dots, p_s)$-partitionable if there exists a partition of $V$ into…

We consider two problems for a directed graph $G$, which we show to be closely related. The first one is to find $k$ edge-disjoint forests in $G$ of maximal size such that the indegree of each vertex in these forests is at most $k$. We…

An equitable $(t,k,d)$-tree-coloring of a graph $G$ is a coloring to vertices of $G$ such that the sizes of any two color classes differ by at most one and the subgraph induced by each color class is a forest of maximum degree at most $k$…

We prove that for any positive integer $k$, the edges of any graph whose fractional arboricity is at most $k + 1/(3k+2)$ can be decomposed into $k$ forests and a matching.

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…