Related papers: Short cycle covers on cubic graphs using chosen 2-…

A matching $M$ in a graph $G$ is uniquely restricted if no other matching in $G$ covers the same set of vertices. We prove that any connected subcubic graph with $n$ vertices and girth at least $5$ contains a uniquely restricted matching of…

A bisection of a graph is a bipartition of its vertex set such that the two resulting parts differ in size by at most 1, and its size is the number of edges that connect vertices in the two parts. The perfect matching condition and…

Given a non-decreasing sequence S = (s 1,s 2,. .. ,s k) of positive integers, an S-packing edge-coloring of a graph G is a partition of the edge set of G into k subsets {X 1 ,X 2,. .. ,X k } such that for each 1 $\le$ i $\le$ k, the…

The perfect matching index of a cubic graph $G$, denoted by $\pi(G)$, is the smallest number of perfect matchings that cover all the edges of $G$. According to the Berge-Fulkerson conjecture, $\pi(G)\le5$ for every bridgeless cubic…

We say that a graph $G$ is $(2,m)$-linked if, for any distinct vertices $a_1,\ldots, a_m, b_1,b_2$ in $G$, there exist vertex disjoint connected subgraphs $A,B$ of $G$ such that $\{a_1, \ldots, a_m\}$ is contained in $A$ and $\{b_1,b_2\}$…

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…

The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all vertices of the graph. It is known that the cover time of any finite connected $n$-vertex graph is at least…

A planar 3-connected graph $G$ is called \emph{essentially $4$-connected} if, for every 3-separator $S$, at least one of the two components of $G-S$ is an isolated vertex. Jackson and Wormald proved that the length $\mathop{\rm…

If $G$ is a claw-free hamiltonian graph of order $n$ and maximum degree $\Delta$ with $\Delta\geq 24$, then $G$ has cycles of at least $\min\left\{ n,\left\lceil\frac{3}{2}\Delta\right\rceil\right\}-2$ many different lengths.

The square of a graph $G$, denoted $G^2$, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most $2$. Thomassen [12] showed that $\chi(G^2) \leq 7$ if $G$ is a subcubic planar…

One way to certify that a graph does not contain an induced cycle of length six is to provide a partition of its vertex set into (i) a stable set, and (ii) a graph containing no stable set of size three and no induced matching of size two.…

Let $G$ be a cubic graph admitting a $2$-factor consisting of exactly two odd circuits, and let the complementary $1$-factor contain precisely three spokes (along with an arbitrary number of chords). We show that four perfect matchings can…

Let $G$ be a simple graph of order $n$ and size $m$. The edge covering of $G$ is a set of edges such that every vertex of $G$ is incident to at least one edge of the set. The edge cover polynomial of $G$ is the polynomial…

Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length $2n+1$ that have exactly $n$ or $n+1$ entries equal to 1, with an edge between any two vertices for which the corresponding bitstrings differ…

A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L. The weight of a cycle cover of an edge-weighted graph is…

Given a bridgeless graph G, the well-known cycle double cover conjecture posits that there is a list of cycles of G, such that every edge appears in exactly two cycles. In this paper, we prove the cycle double cover conjecture. More…

Let $G$ be a graph, and $v\in V(G)$ and $S\subseteq V(G)\backslash v$ of size at least $k$. An important result on graph connectivity due to Perfect states that, if $v$ and $S$ are $k$-linked, then a $(k-1)$-link between a vertex $v$ and…

Let $G$ be a graph. A total dominating set in a graph $G$ is a set $S$ of vertices of $G$ such that every vertex in $G$ is adjacent to a vertex in $S$. Recently, the following question was proposed: "Is it true that every connected cubic…

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…

A cycle $C$ of length $k$ in graph $G$ is extendable if there is another cycle $C'$ in $G$ with $V(C) \subset V(C')$ and length $k+1$. A graph is cycle extendable if every non-Hamiltonian cycle is extendable. In 1990 Hendry conjectured that…