The Hamilton-Waterloo problem asks for which s and r the complete graph K n can be decomposed into s copies of a given 2-factor F 1 and r copies of a given 2-factor F 2 (and one copy of a 1-factor if n is even). In this paper we generalize the problem to complete equipartite graphs K (n:m) and show that K (xyzw:m) can be decomposed into s copies of a 2-factor consisting of cycles of length xzm; and r copies of a 2-factor consisting of cycles of length yzm, whenever m is odd, s, r = 1, gcd(x, z) = gcd(y, z) = 1 and xyz = 0 (mod 4). We also give some more general constructions where the cycles in a given two factor may have different lengths. We use these constructions to find solutions to the Hamilton-Waterloo problem for complete graphs.
A partial Steiner triple system of order
n is sequenceable if there is a sequence of length
n of its distinct points such that no proper segment of the sequence is a union of point‐disjoint blocks. We prove that if a partial Steiner triple system has at most three point‐disjoint blocks, then it is sequenceable.
We present a necessary and sufficient condition for the singularity of circulant matrices associated with directed weighted cycles. This condition is simple and independent of the order of matrices from a complexity point of view. We give explicit and simple formulas for the Drazin inverse of these circulant matrices. We also provide a Bjerhammar-type condition for the Drazin inverse.
The Hamilton-Waterloo problem asks for a decomposition of the complete graph into r copies of a 2-factor F 1 and s copies of a 2-factor F 2 such that r + s = v−1 2 . If F 1 consists of m-cycles and F 2 consists of n cycles, then we call such a decomposition a (m, n)−HWP(v; r, s). The goal is to find a decomposition for every possible pair (r, s). In this paper, we show that for odd x and y, there is a (2 k x, y) − HWP(vm; r, s) if gcd(x, y) ≥ 3, m ≥ 3, and both x and y divide v, except possibly when 1 ∈ {r, s}.
This paper considers an infection spreading in a graph; a vertex gets infected if at least two of its neighbors are infected. The $P_3$-hull number is the minimum size of a vertex set that eventually infects the whole graph.
In the specific case of the Kneser graph $K(n,k)$, with $n\ge 2k+1$, an infection spreading on the family of $k$-sets of an $n$-set is considered. A set is infected whenever two sets disjoint from it are infected. We compute the exact value of the $P_3$-hull number of $K(n,k)$ for $n>2k+1$. For $n = 2k+1$, using graph homomorphisms from the Knesser graph to the Hypercube, we give lower and upper bounds.
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