相关论文: An extension of a cubic 2-connected plane graph G …

We study the existence of hamiltonian cycles in plane cubic graphs G having a facial 2-factor Q. Thus hamiltonicity in G is transformed into the existence of a (quasi) spanning tree of faces in the contraction G/Q. In particular, we study…

We study biplane graphs drawn on a finite point set $S$ in the plane in general position. This is the family of geometric graphs whose vertex set is $S$ and which can be decomposed into two plane graphs. We show that every sufficiently…

A 2-factor of a graph $G$ is a 2-regular spanning subgraph of $G$. We present a survey summarising results on the structure of 2-factors in regular graphs, as achieved by various researchers in recent years.

We call a set $\mathcal S$ of graphs an "even subdivison-factor" of a cubic graph $G$ if $G$ contains a spanning subgraph $H$ such that every component of $H$ has an even number of vertices and is a subdivision of an element of $\mathcal…

Let $P$ be a cubic $3$-connected bipartite plane graph which has a $2$-factor which consists only of facial $4$-cycles, and suppose that $P^{*}$ is the dual graph. We show that $P$ has at least $3^{\frac{2|P^{*}|}{\Delta^{2}{(P^{*})}}}$…

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…

Let $G$ be a simple graph of order $n$. The double vertex graph $F_2(G)$ of $G$ is the graph whose vertices are the $2$-subsets of $V(G)$, where two vertices are adjacent in $F_2(G)$ if their symmetric difference is a pair of adjacent…

Given a set $P$ of $n$ points in the plane, we solve the problems of constructing a geometric planar graph spanning $P$ 1) of minimum degree 2, and 2) which is 2-edge connected, respectively, and has max edge length bounded by a factor of 2…

The bipartite-hole-number of a graph $G$, denoted by $\widetilde{\alpha}(G)$, is the minimum number $k$ such that there exist positive integers $s$ and $t$ with $s+t=k+1$ with the property that for any two disjoint sets $A,B\subseteq V(G)$…

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…

On the basis of recent results on hamiltonicity and hamiltonian connectedness in the square of a 2-block, we determine the most general block-cutvertex structure a graph $G$ may have in order to guarantee that $G^2$ is hamiltonian,…

The bipartite-hole-number of a graph $G$, denoted as $\widetilde{\alpha}(G)$, is the minimum number $k$ such that there exist positive integers $s$ and $t$ with $s+t=k+1$ with the property that for any two disjoint sets $A,B\subseteq V(G)$…

The complete double vertex graph $M_2(G)$ of $G$ is defined as the graph whose vertices are the $2$-multisubsets of $V(G)$, and two of such vertices are adjacent in $M_2(G)$ if their symmetric difference (as multisets) is a pair of adjacent…

The $k$th power of a graph $G$, denoted $G^k$, has the same vertex set as $G$, and two vertices are adjacent in $G^k$ if and only if there exists a path between them in $G$ of length at most $k$. A $K_r$-factor in a graph is a spanning…

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…

For any positive integer $s$, a $[2,2s]$-factor in a graph $G$ is a connected even factor with maximum degree at most $2s$. We prove that if every induced $S(K_{1, 2s+1})$ in a graph $G$ has at least 3 edges in a block of degree at most…

For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane closed curve whose chord diagram contains the given chord diagram as a…

A graph $G$ admiting a $2$-factor is \textit{pseudo $2$-factor isomorphic} if the parity of the number of cycles in all its $2$-factors is the same. In [M. Abreu, A.A. Diwan, B. Jackson, D. Labbate and J. Sheehan. Pseudo $2$-factor…

Given a connected outerplanar graph G of pathwidth p, we give an algorithm to add edges to G to get a supergraph of G, which is 2-vertex-connected, outerplanar and of pathwidth O(p). This settles an open problem raised by Biedl, in the…

A graph is called 1-planar if there exists its drawing in the plane such that each edge is crossed at most once. In this paper, we study 1-planar graph joins. We prove that the join $G+H$ is 1-planar if and only if the pair $[G,H]$ is…