Internally Fair Factorizations and Internally Fair Holey Factorizations with Prescribed Regularity

Let $G$ be a multipartite multigraph without loops. Then $G$ is said to be internally fair if its edges are shared as evenly as possible among all pairs of its partite sets. An internally fair factorization of $G$ is an edge-decomposition of $G$ into internally fair regular spanning subgraphs. A holey factor of $G$ is a regular subgraph spanning all vertices but one partite set. An internally fair holey factorization is an edge-decomposition of $G$ into internally fair holey factors. In this paper, we settle the existence of internally fair (respectively, internally fair holey) factorizations of the complete equipartite multigraph into factors (respectively, holey factors) with prescribed regularity.

On 4-regular 4-connected bipancyclic subgraphs of hypercubes

We consider the problem of determining the possible orders for [Formula: see text]-regular, [Formula: see text]-connected and bipancyclic subgraphs of the hypercube [Formula: see text] For [Formula: see text] and [Formula: see text] the solution to the problem is known. In this paper, we solve the problem for [Formula: see text] by proving that [Formula: see text] has a 4-regular, 4-connected and bipancyclic subgraph on [Formula: see text] vertices if and only if [Formula: see text] or [Formula: see text] is an even integer such that [Formula: see text] Further, by improving a result of Ramras, we prove that a [Formula: see text]-regular subgraph of [Formula: see text] is either isomorphic to [Formula: see text] or has at least [Formula: see text] vertices. We also improve a result of Mane and Waphare regarding the existence of a [Formula: see text]-regular, [Formula: see text]-connected and bipancyclic subgraph of [Formula: see text] Some applications of our results are given.

The Firstk-Regular Subgraph is Large

Let$(G_m)_{0\le m\le \binom{n}{2}}$be the random graph process starting from the empty graph on vertex set [n] and with a random edge added in each step. Letmkdenote the minimum integer such thatGmkcontains ak-regular subgraph. We prove that for all sufficiently largek, there exist two constants εk≥ σk> 0, with εk→ 0 ask→ ∞, such that asymptotically almost surely anyk-regular subgraph ofGmkhas size between (1 − εk)|${\mathcal C}_k$| and (1 − σk)|${\mathcal C}_k$|, where${\mathcal C}_k$denotes thek-core ofGmk.

An O(n) Time Algorithm to Find Cubic Subgraph in a Halin Graph

The 3-Regular Subgraph Problem is: Given a graph G = (V, E), can we find a subgraph H = (V, E) in G such that for each vertex u in V, , where is the degree of u in H? This problem is an NP-complete problem for general graphs. In this paper, we design an O(n) time algorithm to solve The 3-Regular Subgraph Problem for a Halin graph H, where n is the number of vertices of H. Given a Halin graph H, if there is a cubic subgraph G in H, then our algorithm will find G and give an answer Yes, otherwise our algorithm will give an answer No. We also prove the correctness of this algorithm.

(k+1)-Cores Have k-Factors

We prove that the threshold for the appearance of a k-regular subgraph in Gn,p is at most the threshold for the appearance of a non-empty (k+1)-core. This improves a result of Pralat, Verstraete and Wormald [5] and proves a conjecture of Bollobás, Kim and Verstraete [3].