Related papers: The Frank number and nowhere-zero flows on graphs

We prove the existence of a function f: N^2 -> N such that for all p,k in N every (k(p-3) + 14p+14) - connected graph either has k disjoint K_p minors or contains a set of at most f(p,k) vertices whose deletion kills all its K_p minors. For…

For a given snark G and edge e of G, we can form a cubic graph G_e using an operation we call "edge subtraction". The number of 3-edge-colourings of G_e is 18 * \psi(G,e) for some nonnegative integer \psi(G,e). Given snarks G_1 and G_2, we…

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

Snarks are $2$-connected cubic graphs that do not admit a proper $3$-edge-coloring. For a cubic graph $G$, its resistance $r(G)$ is the minimum number of edges whose removal results in a $3$-edge-colorable graph, while its flow resistance…

We estimate the minimum number of vertices of a cubic graph with given oddness and cyclic connectivity. We prove that a bridgeless cubic graph $G$ with oddness $\omega(G)$ other than the Petersen graph has at least $5.41\cdot\omega(G)$…

A graph $G$ is called collapsible if for every even subset $R\subseteq V(G)$, there is a spanning connected subgraph $H$ of $G$ such that $R$ is the set of vertices of odd degree in $H$. A graph is the reduction of $G$ if it is obtained…

We disprove a conjecture of Frank stating that each weakly 2k-connected has a k-vertex-connected orientation. For k at least 3, we also prove that the problem of deciding whether a graph has a k-vertex-connected orientation is NP-complete.

The frustration index of a signed graph is defined as the minimum number of negative edges among all switching-equivalent signatures. This can be regarded as a generalization of the classical \textsc{Max-Cut} problem in graphs, as the…

A path in an edge-coloured graph is called \emph{rainbow path} if its edges receive pairwise distinct colours. An edge-coloured graph is said to be \emph{rainbow connected} if any two distinct vertices of the graph are connected by a…

A graph $G$ is called degree-truncated $k$-choosable if for every list assignment $L$ with $|L(v)| \ge \min\{d_G(v), k\}$ for each vertex $v$, $G$ is $L$-colourable. Richter asked whether every 3-connected non-complete planar graph is…

Given a simple undirected graph $G$ and a positive integer $k$, the $k$-forcing number of $G$, denoted $F_k(G)$, is the minimum number of vertices that need to be initially colored so that all vertices eventually become colored during the…

A simple graph $G$ is \textit{k-ordered} (respectively, \textit{k-ordered hamiltonian}), if for any sequence of $k$ distinct vertices $v_1, ..., v_k$ of $G$ there exists a cycle (respectively, hamiltonian cycle) in $G$ containing these $k$…

The cut-rank of a set $X$ of vertices in a graph $G$ is defined as the rank of the $ X \times (V(G)\setminus X)$ matrix over the binary field whose $(i,j)$-entry is $1$ if the vertex $i$ in $X$ is adjacent to the vertex $j$ in…

A connected graph $G$ with at least two vertices is matching covered if each of its edges lies in a perfect matching. A matching covered graph is minimal if the removal of any edge results in a graph that is no longer matching covered. An…

An orientation of $G$ is a digraph obtained from $G$ by replacing each edge by exactly one of two possible arcs with the same endpoints. We call an orientation \emph{proper} if neighbouring vertices have different in-degrees. The proper…

{ An edge $e$ in a matching covered graph $G$ is {\em removable} if $G-e$ is matching covered, which was introduced by Lov\'asz and Plummer in connection with ear decompositions of matching covered graphs. A {\it brick}} is a non-bipartite…

A forcing set for a perfect matching of a graph is defined as a subset of the edges of that perfect matching such that there exists a unique perfect matching containing it. A complete forcing set for a graph is a subset of its edges, such…

A proper orientation $D$ of an undirected graph $G$ is an orientation of $G$ such that $d_D^+(u)\not=d_D^+(v)$ for any edge $uv\in E(G)$. Denote the proper orientation number $\vec{\chi}(G)$ of an undirected graph $G$ as the minimum…

Amos et al. (Discrete Appl. Math. 181 (2015) 1-10) introduced the notion of the $k$-forcing number of graph for a positive integer $k$ as the generalization of the zero forcing number of a graph. The $k$-forcing number of a simple graph…

For two integers $r\geq 2$ and $h\geq 0$, the \emph{$h$-extra $r$-component connectivity} $\kappa^h_r(G)$ of a graph $G$ is defined to be the minimum size of a subset of vertices whose removal disconnects $G$, and there are at least $r$…