Related papers: A short proof of Seymour's 6-flow theorem
A cornerstone theorem in the Graph Minors series of Robertson and Seymour is the result that every graph $G$ with no minor isomorphic to a fixed graph $H$ has a certain structure. The structure can then be exploited to deduce far-reaching…
We say a graph $G$ has a Hamiltonian path if it has a path containing all vertices of $G$. For a graph $G$, let $\sigma_2(G)$ denote the minimum degree sum of two nonadjacent vertices of $G$; restrictions on $\sigma_2(G)$ are known as…
We show that every 3--manifold admits a Heegaard diagram in which a truncated version of Heegaard Floer homology (when the holomorpic disks pass through the basepoints at most once) can be computed combinatorially.
In this paper, we are motivated by the conjectures proposed by C.~Bender \textit{et al.}, \cite{C} in 2024. We have settled the first two conjectures negatively by providing a counter example in \cite{KTJ}, whereas in this paper, we prove…
The paper is devoted to the study of topological properties, structure and classification of Morse flows with fixed points on the boundary of three-dimensional manifolds. We construct a complete topological invariant of a Morse flow,…
A vertex in a directed graph is said to have a large second neighborhood if it has at least as many second out-neighbors as out-neighbors. The Second Neighborhood Conjecture, first stated by Seymour, asserts that there is a vertex having a…
In 1980, Thomassen stated his weak linkage conjecture: for an odd positive integer k, if a graph G is k-edge-connected, then, for any collection of k pairs of vertices {s_1,t_1}, ..., {s_k,t_k} in G, not necessarily distinct, there are…
A graph is pseudo 2-factor isomorphic if all of its 2-factors have the same parity of number of cycles. Abreu et al. [J. Comb. Theory, Ser. B. 98 (2008) 432--442] conjectured that $K_{3,3}$, the Heawood graph and the Pappus graph are the…
For any pair of edges $e,f$ of a graph $G$, we say that {\em $e,f$ are $P_3$-connected in $G$} if there exists a sequence of edges $e=e_0,e_1,\ldots, e_k=f$ such that $e_i$ and $e_{i+1}$ are two edges of an induced $3$-vertex path in $G$…
Let $G$ and $H$ be simple 3-connected graphs such that $G$ has an $H$-minor. An edge $e$ in $G$ is called {\it $H$-deletable} if $G\backslash e$ is 3-connected and has an $H$-minor. The main result in this paper establishes that, if $G$ has…
A graph $G$ is pseudo 2-factor isomorphic if the parity of the number of cycles in a 2-factor is the same for all 2-factors of $G$. Abreu et al. conjectured that $K_{3,3}$, the Heawood graph and the Pappus graph are the only essentially…
In this paper we provide some sufficient conditions for the existence of an odd or even cycle that passing a given vertex or an edge in $2$-connected or $2$-edge connected graphs. We provide some similar conditions for the existence of an…
We attempt to generalize a theorem of Nash-Williams stating that a graph has a $k$-arc-connected orientation if and only if it is $2k$-edge-connected. In a strongly connected digraph we call an arc {\it deletable} if its deletion leaves a…
The number of nowhere zero Z_Q flows on a graph G can be shown to be a polynomial in Q, defining the flow polynomial \Phi_G(Q). According to Tutte's five-flow conjecture, \Phi_G(5) > 0 for any bridgeless G.A conjecture by Welsh that…
We show that for any claw-free graph $G$ and any graph $H$, $\gamma(G\square H)\geq \frac{2}{3}\gamma(G)\gamma(H)$, where $\gamma(G)$ is the domination number of $G$.
Seymour's Second-Neighborhood Conjecture states that every directed graph whose underlying graph is simple has at least one vertex $v$ such that the number of vertices of out-distance $2$ from $v$ is at least as large as the number of…
Let $\mathcal{G}$ be the class of all graphs with no induced four-edge path or four-edge antipath. Hayward and Nastos \cite{MS} conjectured that every prime graph in $\mathcal{G}$ not isomorphic to the cycle of length five is either a split…
We announce results about flat (linkless) embeddings of graphs in 3-space. A piecewise-linear embedding of a graph in 3-space is called {\it flat} if every circuit of the graph bounds a disk disjoint from the rest of the graph. We have…
An arc of a graph is an oriented edge and a 3-arc is a 4-tuple $(v,u,x,y)$ of vertices such that both $(v,u,x)$ and $(u,x,y)$ are paths of length two. The 3-arc graph of a graph $G$ is defined to have vertices the arcs of $G$ such that two…
The Second Neighborhood Conjecture of Seymour asserts that every oriented graph contains a vertex~$v$ satisfying $|\Npp(v)|\ge|\Np(v)|$. We introduce \emph{Pisa graphs} -- strongly connected oriented graphs~$D$ with $\Delta(D)=\max_{v\in…