Related papers: On Barnette's Conjecture and $H^{+-}$ property

We find Dirac-type sufficient conditions for a hypergraph $\mathcal H$ with few edges to be hamiltonian. We also show that these conditions provide that $\mathcal H$ is {\em super-pancyclic}, i.e., for each $A \subseteq V(\mathcal H)$ with…

We use a randomised embedding method to prove that for all \alpha>0 any sufficiently large oriented graph G with minimum in-degree and out-degree \delta^+(G),\delta^-(G)\geq (3/8+\alpha)|G| contains every possible orientation of a Hamilton…

For a collection $\mathbf{G}=\{G_1,\dots, G_s\}$ of not necessarily distinct graphs on the same vertex set $V$, a graph $H$ with vertices in $V$ is a $\mathbf{G}$-transversal if there exists a bijection $\phi:E(H)\rightarrow [s]$ such that…

Let $\mathcal{G}(k)$ denote the set of connected $k$-regular graphs $G$, $k\geq2$, where the number of vertices at distance 2 from any vertex in $G$ does not exceed $k$. Asratian (2006) showed (using other terminology) that a graph…

Scott proved in 1997 that for any tree $T$, every graph with bounded clique number which does not contain any subdivision of $T$ as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if $T$ is…

The Erdos-Hajnal Conjecture asserts that for every graph H there is a constant c > 0 such that every graph G that does not contain H as an induced subgraph has a clique or stable set of cardinality at least |G|^c. In this paper, we prove a…

A question at the intersection of Barnette's Hamiltonicity and Neumann-Lara's dicoloring conjecture is: Can every Eulerian oriented planar graph be vertex-partitioned into two acyclic sets? A CAI-partition of an undirected/oriented graph is…

A celebrated conjecture of Sidorenko and Erd\H{o}s-Simonovits states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge…

Sidorenko's conjecture states that for every bipartite graph $H$ on $\{1,\cdots,k\}$, $\int \prod_{(i,j)\in E(H)} h(x_i, y_j) d\mu^{|V(H)|} \ge \left( \int h(x,y) \,d\mu^2 \right)^{|E(H)|}$ holds, where $\mu$ is the Lebesgue measure on…

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic edge coloring conjecture by Fiam{\v{c}}ik (1978) and Alon, Sudakov and Zaks (2001) states that every simple graph…

A graph $G$ is said to be perfectly divisible if for every induced subgraph $H$ of $G$ with at least one edge, the vertex set $V(H)$ can be partitioned into two sets $A, B$ such that $H[A]$ is perfect and $\omega(B) < \omega(H)$. It is easy…

In 1981 Jackson showed that the diregular bipartite tournament (a complete bipartite graph whose edges are oriented so that every vertex has the same in- and outdegree) contains a Hamilton cycle, and conjectured that in fact the edge set of…

Following a problem posed by Lov\'asz in 1969, it is believed that every connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from groups having a $(2,s,3)$-presentation, that…

Barnette's conjecture states that every cubic, bipartite, planar and 3-connected graph is Hamiltonian. Goodey verified Barnette's conjecture for all graphs with faces of size up to 6. We substantially strengthen Goodey's result by proving…

We show that every edge in a 2-edge-connected planar cubic graph is either contained in a 2-edge-cut or is a chord of some cycle that is contained in a 2-factor of the graph. As a consequence, we show that every edge in a cyclically…

We study $M$-alternating Hamilton paths and $M$-alternating Hamilton cycles in a simple connected graph $G$ on $\nu$ vertices with a perfect matching $M$. Let $G$ be a bipartite graph, we prove that if for any two vertices $x$ and $y$ in…

In this paper, we deal with hamiltonicity in planar cubic graphs G having a facial 2-factor Q via (quasi) spanning trees of faces in G/Q and study the algorithmic complexity of finding such (quasi) spanning trees of faces. Moreover, we show…

A perfect matching cut is a perfect matching that is also a cutset, or equivalently a perfect matching containing an even number of edges on every cycle. The corresponding algorithmic problem, Perfect Matching Cut, is known to be…

For a graph $G$ the random $n$-lift of $G$ is obtained by replacing each of its vertices by a set of $n$ vertices, and joining a pair of sets by a random matching whenever the corresponding vertices of $G$ are adjacent. We show that…

An $n$-vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices and it is pancyclic if it contains cycles of all lengths from $3$ up to $n$. A celebrated meta-conjecture of Bondy states that every non-trivial…