Related papers: Decomposing planar graphs into graphs with degree …

Core decomposition is a fundamental operator in network analysis. In this paper, we study the problem of computing distance-generalized core decomposition on a network. A distance-generalized core, also termed $(k, h)$-core, is a maximal…

It is well known that the spectral radius of a tree whose maximum degree is D cannot exceed 2sqrt{D-1}. Similar upper bound holds for arbitrary planar graphs, whose spectral radius cannot exceed sqrt{8D}+10, and more generally, for all…

For a graph $G$, by $\chi_2(G)$ we denote the minimum integer $k$, such that there is a $k$-coloring of the vertices of $G$ in which vertices at distance at most 2 receive distinct colors. Equivalently, $\chi_2(G)$ is the chromatic number…

A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one another edge. In this work we prove that each 1-planar graph of minimum degree at least $3$ contains an edge with degrees of its endvertices of…

In this paper, we study the concept of edge-group choosability of graphs. We say that G is edge k-group choosable if its line graph is k-group choosable. An edge-group choosability version of Vizing conjecture is given. The evidence of our…

For some $k \in \mathbb{Z}_{\geq 0}\cup \infty$, we call a linear forest $k$-bounded if each of its components has at most $k$ edges. We will say a $(k,\ell)$-bounded linear forest decomposition of a graph $G$ is a partition of $E(G)$ into…

The basis number of a graph $G$ is the minimum $k$ such that the cycle space of $G$ is generated by a family of cycles using each edge at most $k$ times. A classical result of Mac Lane states that planar graphs are exactly graphs with basis…

We consider the class F of 2-connected non-planar K_{3,3}-subdivision-free graphs that are embeddable in the projective plane. We show that these graphs admit a unique decomposition as a graph K_5 (the core) where the edges are replaced by…

For an integer $d\geq 1$, the $d$-Cut problem is that of deciding whether a graph has an edge cut in which each vertex is adjacent to at most $d$ vertices on the opposite side of the cut. The $1$-Cut problem is the well-known Matching Cut…

A longest path in a graph is called a detour. It is easy to see that a connected graph of minimum degree at least $2$ and order at least $4$ has at least $4$ detours. We prove that if the number of detours in such a graph of order at least…

A univariate polynomial f over a field is decomposable if it is the composition f = g(h) of two polynomials g and h whose degree is at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and…

Let $G$ be a graph and let $l$ be an integer-valued function on subsets of $V(G)$. The graph $G$ is said to be $l$-partition-connected, if for every partition $P$ of $V(G)$, $e_G(P)\ge \sum_{A\in P} l(A)-l(V(G))$, where $e_G(P)$ denotes the…

We prove a robust contraction decomposition theorem for $H$-minor-free graphs, which states that given an $H$-minor-free graph $G$ and an integer $p$, one can partition in polynomial time the vertices of $G$ into $p$ sets $Z_1,\dots,Z_p$…

Over 50 years ago, Erd\H{o}s and Gallai conjectured that the edges of every graph on $n$ vertices can be decomposed into $O(n)$ cycles and edges. Among other results, Conlon, Fox and Sudakov recently proved that this holds for the random…

Gallai's path decomposition conjecture states that for a connected graph $G$ on $n$ vertices, there exists a path decomposition of size $\lceil \frac{n}{2} \rceil$. The Levi graph of order one, denoted by $L_{1}(m,k)$, is a bipartite graph…

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every $2$-coloring of the…

For planar graphs, we consider the problems of \emph{list edge coloring} and \emph{list total coloring}. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total…

We settle a problem of Dujmovi\'c, Eppstein, Suderman, and Wood by showing that there exists a function $f$ with the property that every planar graph $G$ with maximum degree $d$ admits a drawing with noncrossing straight-line edges, using…

Assume $G$ is a graph and $k$ is a positive integer. Let $f:V(G)\to \mathbb{N}$ be defined as $f(v)=\min\{k,d_G(v)\}$. If $G$ is $f$-choosable, then we say $G$ is degree-truncated $k$-choosable. Answering a question of Richter, it was…

An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If $\M(G)$ denotes the minimum size of an identifying code of a graph $G$, it was…