Related papers: M\'acajov\'a and \v{S}koviera Conjecture on Cubic …
It was conjectured by Mkrtchyan, Petrosyan, and Vardanyan that every graph $G$ with $\Delta(G)-\delta(G) \le 1$ has a maximum matching $M$ such that any two $M$-unsaturated vertices do not share a neighbor. In this note, we confirm the…
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…
We show that no cubic graphs of order 26 have crossing number larger than 9, which proves a conjecture of Ed Pegg Jr and Geoffrey Exoo that the smallest cubic graphs with crossing number 11 have 28 vertices. This result is achieved by first…
It was conjectured by Hoffmann-Ostenhof that the edge set of every connected cubic graph can be decomposed into a spanning tree, a matching and a family of cycles. In this paper, we show that this conjecture holds for traceable cubic…
We conjecture that every graph of minimum degree five with no separating triangles and drawn in the plane with one crossing is 4-colorable. In this paper, we use computer enumeration to show that this conjecture holds for all graphs with at…
The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general, while it is known to be…
In this paper we have shown without assuming the four color theorem of planar graphs that every (bridgeless) cubic planar graph has a three-edge-coloring. This is an old-conjecture due to Tait in the squeal of efforts in settling the…
Circular arc graphs are graphs whose vertices can be represented as arcs on a circle such that any two vertices are adjacent if and only if their corresponding arcs intersect. Proper circular arc graphs are graphs which have a circular arc…
In graph theory, a graceful labeling of a graph with m edges is a labeling of its vertices with a subset of the integers ranging from 0 to m inclusive, such that no two vertices share a label, and each edge is uniquely identified by the…
A graph is {\em perfect} if, in all its induced subgraphs, the size of a largest clique is equal to the chromatic number. Examples of perfect graphs include bipartite graphs, line graphs of bipartite graphs and the complements of such…
We study decomposition into simple arcs (i. e., arcs without self-intersections) for diagrams of knots and spatial graphs. In this paper, it is proved in particular that if no edge of a finite spatial graph $G$ is a knotted loop, then there…
We show that for a sequence of random graphs Brouwer's conjecture holds true with probability tending to one as the number of vertices tends to infinity. Surprisingly, it was found that a similar statement holds true for weighted graphs…
The cycle double cover conjecture states that a graph is bridge-free if and only if there is a family of edge-simple cycles such that each edge is contained in exactly two of them. It was formulated independently by Szekeres (1973) and…
Let v(G) and dom(G) denote the number of vertices and the domination number of a graph G, and let r (G) = dom(G)/v(G)$. Let [x] and ]x[ be the floor and the ceiling of a number x. In 1996 B. Reed conjectured that if G is a cubic graph, then…
Perfect matchings and maximum weight matchings are two fundamental combinatorial structures. We consider the ratio between the maximum weight of a perfect matching and the maximum weight of a general matching. Motivated by the computer…
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 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…
A vertex $v$ of a 2-connected cubic graph $G$ is $\lambda$-matchable if $G$ has a spanning subgraph in which $v$ has degree three whereas every other vertex has degree one, and we let $\lambda(G)$ denote the number of such vertices.…
A beautiful conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same…
An $r$-graph is an $r$-regular graph where every odd set of vertices is connected by at least $r$ edges to the rest of the graph. Seymour conjectured that any $r$-graph is $r+1$-edge-colorable, and also that any $r$-graph contains $2r$…