On matchings and factors of graphs /

In Section 1, we recall the historical sketch of matching and factor theory of graphs, and also introduce some necessary definitions and notation. In Section 2, we present a sufficient condition for the existence of a (g, f)-factor in graphs with the odd-cycle property, which is simpler than that of Lovasz\u27s (g, f)-Factor Theorem. From this, we derive some further results, and we show that (a) every r-regular graph G with the odd-cycle property has a k-factor, where 0 ≤ k ≤ r and k|V(G)| ≡ 0 (mod 2), (b) every graph G with the strong odd-cycle property with k|V(G)|≡ 0 (mod 2) is k-factorable if and only if G is a km-regular graph for some m ≥ 1, and (c) every regular graph of even order with the strong odd-cycle property is of the second class (i.e. the edge chromatic number is Δ). Chvátal [26] presented the following two conjectures that (1) a graph G has a 2-factor if tough(G) ≥ 3/2, and (2) a graph G has a k-factor if k|V(G)| ≡ 0 (mod 2) and tough(G) ≥ k. Enomoto et.al. [32] proved the second conjecture. They also proved the sharpness of the bound on tough(G) that guarantees the existence of a k-factor. This implies that the first conjecture is false. In Section 3, we show that the result of the second conjecture can be improved in some sense, and the first conjecture is also true if the graph considered has the odd-cycle property. Anderson [3] stated that a graph G of even order has a 1-factor if bind(G) ≥ 4/3, and Katerinis and Woodall [48] proved that a graph G of order n has a k-factor if bind(G) ˃ (2k -I)(n - 1)/(k(n - 2) + 3), where k ≥ 2, n ≥ 4k - 6 and kn ≡ 0 (mod 2). In Section 4, we shall present some similar conditions for the existence of [a, b]-factors. In Section 5, we study the existence of [a, b]-parity-factors in a graph, among which we extend some known theorems from 1-factors to {1, 3, ... , 2n - 1}-factors, or from k-factors to [a, b]-parity-factors. Also, extending Petersen\u27s 2-Factorization Theorem, we proved that a graph is [2a, 2b]-even-factorable if and only if it is a [2na, 2nb]-even-graph for some n ≥ 1. Plummer showed that (a) (in [58]) every graph G of even order is k-extendable if tough(G) ˃ k, and (b) (in [59]) every (2k+1)-connected graph G is k-extendable if G is K1,3-free, respectively. In Section 6, we give a counterpart of the former in terms of binding number, and extend the latter from K1,3-free graphs to K1,n-free graphs. Furthermore, we present a result toward the problem, posed by Saito [61] and Plummer [60], of characterizing the graphs that are maximal k-extendable

A Generalization of the Hamilton-Waterloo Problem on Complete Equipartite Graphs

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, $s,r\neq 1$, $\gcd(x,z)=\gcd(y,z)=1$ and $xyz\neq 0 \pmod 4$. 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