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Komlos [Tiling Turan theorems, Combinatorica, 20,2 (2000), 203{218] determined the asymptotically optimal minimum degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph. We show that the minimum degree condition can be relaxed in the sense that we require only a given fraction of vertices to have the prescribed degree.
Quasi-median graphs are a tool commonly used by evolutionary biologists to visualise the evolution of molecular sequences. As with any graph, a quasi-median graph can contain cut vertices, that is, vertices whose removal disconnect the graph. These v
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 Ore-type conditio
A graph $G$ is $k$-path-coverable if its vertex set $V(G)$ can be covered by $k$ or fewer vertex disjoint paths. In this paper, using the $Q$-index of a connected graph $G$, we present a tight sufficient condition for $G$ with fixed minimum degree and large order to be $k$-path-coverable.
We study the generating function of descent numbers for the permutations with descent pairs of prescribed parities, the distribution of which turns out to be a refinement of median Genocchi numbers. We prove the $gamma$-positivity for the polynomial
In this paper a closed form expression for the number of tilings of an $ntimes n$ square border with $1times 1$ and $2times1$ cuisenaire rods is proved using a transition matrix approach. This problem is then generalised to $mtimes n$ rectangular bor