ABSTRACTA subgraph of a multigraph is overfull if Analogous to the Overfull Conjecture proposed by Chetwynd and Hilton in 1986, Stiebitz et al. formed the multigraph version of the conjecture as follows: Let be a multigraph with maximum multiplicity and maximum degree . Then has chromatic index if and only if contains no overfull subgraph. In this paper, we prove the following three results toward the Multigraph Overfull Conjecture for sufficiently large and even , where . (1) If is ‐regular with , then has a 1‐factorization. This result also settles a conjecture of the first author and Tipnis from 2001 up to a constant error in the lower bound of . (2) If contains an overfull subgraph and , then , where is the fractional chromatic index of . (3) If the minimum degree of is at least for any and contains no overfull subgraph, then . The proof is based on the decomposition of multigraphs into simple graphs and we prove a slightly weaker version of a conjecture due to the first author and Tipnis from 1991 on decomposing a multigraph into constrained simple graphs. The result is also of independent interest.

Let f: {1, ..., n}→{1, ..., n} be a function (not necessarily one-to-one). An f-derangement is a permutation g:{1,...,n}→{1,...,n} such that g(i) ≠ f(i) for each i = 1, ..., n. When f is itself a permutation, this is a standard derangement. We examine properties of f-derangements, and show that when we fix the maximum number of preimages for any item under f, the fraction of permutations that are f-derangements tends to 1/e for large n, regardless of the choice of f. We then use this result to analyze a heuristic method to decompose bipartite graphs into paths of length 5

Let G $G$ be a simple graph with maximum degree Δ ( G ) ${\rm{\Delta }}(G)$ . A subgraph H $H$ of G $G$ is overfull if ∣ E ( H ) ∣ > Δ ( G ) ⌊ ∣ V ( H ) ∣ ∕ 2 ⌋ . $| E(H)| \gt {\rm{\Delta }}(G)\lfloor | V(H)| \unicode{x02215}2\rfloor .$ Chetwynd and Hilton in 1986 conjectured that a graph G $G$ with Δ ( G ) > ∣ V ( G ) ∣ ∕ 3 ${\rm{\Delta }}(G)\gt | V(G)| \unicode{x02215}3$ has chromatic index Δ ( G ) ${\rm{\Delta }}(G)$ if and only if G $G$ contains no overfull subgraph. The best previous results supporting this conjecture have been obtained for regular graphs. For example, Perković and Reed verified the conjecture for large regular graphs G $G$ with degree arbitrarily close to ∣ V ( G ) ∣ ∕ 2 $| V(G)| \unicode{x02215}2$ . We provide a similar result for general graphs asymptotically, showing that for any given 0 < ϵ < 1 $0\lt \epsilon \lt 1$ , there exists a positive integer n 0 ${n}_{0}$ such that the following statement holds: if G $G$ is a graph on 2 n ≥ n 0 $2n\ge {n}_{0}$ vertices with minimum degree at least ( 1 + ϵ ) n $(1+\epsilon )n$ , then G $G$ has chromatic index Δ ( G ) ${\rm{\Delta }}(G)$ if and only if G $G$ contains no overfull subgraph.

The 1-Factorization Conjecture states that any regular graph of even order n in which the vertex degrees are at least n2 has a factorization of its edges into perfect matchings. Chetwynd and Hilton [1], [3] and independently Niessen and Volkmann [11] verified this conjecture when the vertex degrees are at least .83n. This is equivalent to saying that any such regular graph G has chromatic index χ′(G) equal to its maximum degree Δ(G). Using that result, it was shown in [14] that for any graph G (not necessarily regular) of even order n and minimum degree at least .882n, χ′(G)=Δ(G) if and only if there is no vertex v for which the number of edges of G−v surpasses Δ(G)⋅(n2−1). Recently, Csaba, Kühn, Lo, Osthus and Treglown [6] verified that the 1-Factorization Conjecture holds for all graphs with sufficiently large n. Using this result, we show that for any graph with even order n sufficiently large and minimum degree at least 2n3,χ′(G)=Δ(G) if and only if no vertex-deleted subgraph has more than Δ(G)⋅(n2−1) edges. These results are presented in the context of the more general Overfull Conjecture.

Let G be a 4k-regular graph with k >= 2. We show that G can be decomposed into k 4-regular spanning subgraphs G(1), G(2), ..., G(k), each of which does not contain an induced subgraph that is isomorphic to K-5 or K-5 - e. We then use a result of Heinrich et al. [J. Graph Theory 31 (1999) 135-143] which provides a triangle-free Euler tour in each of G(1), G(2), ..., G(k) to show that G has a triangle-free Euler tour. In the case when m is even, our results imply a result by Oksimets thesis, limed University, Umea, 2003] which states that every connected 2m-regular graph G with m >= 2 and vertical bar E(G)vertical bar divisible by 3 can be decomposed into paths of length 3.

Let G be a forest with n edges. El-Zanati conjectures that G necessarily decomposes every 2n-regular graph and every n-regular bipartite graph. We confirm these conjectures in the case when G consists of two stars.

A double-star is a tree with exactly two vertices of degree greater than 1. If T is a double-star where the two vertices of degree greater than one have degrees k(1) + 1 and k(2) + 1, then T is denoted by S-k1,S- (k2). In this note, we show that every double-star with n edges decomposes every 2n-regular graph. We also show that the double-star S-k,S- k-1 decomposes every 2k-regular graph that contains a perfect matching.

Let G denote a multigraph with edge set E(G), let µ(G) denote the maximum edge multiplicity in G, and let Pk denote the path on k vertices. Heinrich et al.(1999) showed that P4 decomposes a connected 4-regular graph G if and only if |E(G)| is divisible by 3. We show that P4 decomposes a connected 4-regular multigraph G with µ(G) ≤2 if and only if no 3 vertices of G induce more than 4 edges and |E(G)| is divisible by 3. Oksimets (2003) proved that for all integers k ≥3, P4 decomposes a connected 2k-regular graph G if and only if |E(G)| is divisible by 3. We prove that for all integers k ≥2, the problem of determining if P4 decomposes a (2k + 1)-regular graph is NP-Complete. El-Zanati et al.(2014) showed that for all integers k ≥1, every 6k-regular multigraph with µ(G) ≤2k has a P4-decomposition. We show that unless P = NP, this result is best possible with respect to µ(G) by proving that for all integers k ≥3 the problem of determining if P4 decomposes a 2k-regular multigraph with µ(G) ≤⌊2k / 3 ⌋+ 1 is NP-Complete.

It is known that P4, the path with 3 edges, decomposes every 6-regular simple graph. It is also known that P4 decomposes the multigraph obtained by doubling each edge of a cubic graph. We show that P4 decomposes every 6-regular multigraph with edge multiplicity at most 2. This in turn implies that P4 decomposes every 6k-regular multigraph with edge multiplicity at most 2k. We also investigate decompositions of certain 2n-regular multigraphs with edge multiplicity at most 2 into double-stars with n edges.

Let D be a directed graph of order n. An anti-directed (hamiltonian) cycle H in D is a (hamiltonian) cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. In this paper we give sufficient conditions for the existence of anti-directed hamiltonian cycles. Specifically, we prove that a directed graph D of even order n with minimum indegree and outdegree greater than \({\frac{1}{2}n + 7\sqrt{n}/3}\) contains an anti-directed hamiltonian cycle. In addition, we show that D contains anti-directed cycles of all possible (even) lengths when n is sufficiently large and has minimum in- and out-degree at least \({(1/2+ \epsilon)n}\) for any \({\epsilon > 0}\) .

The integer round-up φ(G) of the fractional chromatic index yields the standard lower bound for the chromatic index of a multigraph G. We show that if G has even order n, then the chromatic index exceeds φ(G) by at most max{log32 n, 1 + n/30}. More generally, we show that for any real b,23 ⩽ b < 1, the chromatic index of G exceeds φ(G) by at most max{log1/b n, 1 + n(1 − b)/10}. This is used to show that for n sufficiently large, χ(G) ⩽ φ(G) + 1 + √n 1n n/10. Thus the difference between the chromatic index and its lower bound φ(G) is eventually sublinear; that is, for any real c > 0, there exists a positive integer N such that χ(G) − φ(G) < cn for any multigraph G with order n > N.

Let D be a directed graph with vertex set V and order n. An anti-directed hamiltonian cycle H in D is a hamiltonian cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. An anti-directed 2-factor in D is a vertex-disjoint collection of anti-directed cycles in D that span V. It was proved in [3] that if the indegree and the outdegree of each vertex of D is greater than (9/16)n then D contains an anti-directed hamilton cycle. In this paper we prove that given a directed graph D, the problem of determining whether D has an anti-directed 2-factor is NP-complete, and we use a proof technique similar to the one used in [3] to prove that if the indegree and the outdegree of each vertex of D is greater than (24/46)n then D contains an anti-directed 2-factor.

Let $D$ be a directed graph of order $n$. An anti-directed Hamilton cycle $H$ in $D$ is a Hamilton cycle in the graph underlying $D$ such that no pair of consecutive arcs in $H$ form a directed path in $D$. We prove that if $D$ is a directed graph with even order $n$ and if the indegree and the outdegree of each vertex of $D$ is at least ${2\over 3}n$ then $D$ contains an anti-directed Hamilton cycle. This improves a bound of Grant. Let $V(D) = P \cup Q$ be a partition of $V(D)$. A $(P,Q)$ vertex-oriented Hamilton cycle in $D$ is a Hamilton cycle $H$ in the graph underlying $D$ such that for each $v \in P$, consecutive arcs of $H$ incident on $v$ do not form a directed path in $D$, and, for each $v \in Q$, consecutive arcs of $H$ incident on $v$ form a directed path in $D$. We give sufficient conditions for the existence of a $(P,Q)$ vertex-oriented Hamilton cycle in $D$ for the cases when $|P| \geq {2\over 3}n$ and when ${1\over 3}n \leq |P| \leq {2\over 3}n$. This sharpens a bound given by Badheka et al.

Let G be a connected multigraph with an even number of edges and suppose that the degree of each vertex of G is even. Let mu(uv, G) denote the multiplicity of edge (u, v) in G. It is well known that we can obtain a halving of G into two halves G, and G(2), i.e. that G can be decomposed into multigraphs G(1) and G(2), where for each vertex v, deg(v, G(1)) = deg(v, G(2)) = 1/2deg(v, G). It is also easy to see that if the edges with odd multiplicity in G induce no components with an odd number of edges then we can obtain such a halving of G into two halves G, and G2 that is well-spread, i.e. for each edge (u,v) of G, vertical bar mu(uv,G(1)) - mu(uv,G(2))vertical bar <= 1. We show that if G is a Delta-regular multigraph with an even number of vertices and with A being even, then even if the edges with odd multiplicity in G induce components with an odd number of edges, we can still obtain a well-spread halving of G provided that we allow the addition/removal of a Hamilton cycle to/from G. We give an application of this result to obtaining sports schedules such that multiple encounters between teams are well-spread throughout the season.

Let ${\cal F}$ be a 1‐factorization of the complete uniform hypergraph ${\cal G}={K_{rn}^{(r)}}$ with $r \geq 2$ and $n\geq 3$. We show that there exists a 1‐factor of ${\cal G}$ whose edges belong to n different 1‐factors in ${\cal F}$. Such a 1‐factor is called a “rainbow” 1‐factor or an “orthogonal” 1‐factor. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 487–490, 2007

Chetwynd and Hilton showed that any regular graph G of even order n which has relatively high degree $\Delta (G)\,\ge\,((\sqrt{7}- 1)/2)\, n$ has a 1-factorization. This is equivalent to saying that under these conditions G has chromatic index equal to its maximum degree $\Delta(G)$. Using this result, we show that any (not necessarily regular) graph G of even order n that has sufficiently high minimum degree $\delta(G)\,\ge\,(\sqrt{7}/3)\,n$ has chromatic index equal to its maximum degree providing that G does not contain an “overfull” subgraph, that is, a subgraph which trivially forces the chromatic index to be more than the maximum degree. This result thus verifies the Overfull Conjecture for graphs of even order and sufficiently high minimum degree. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 73–80, 2004

Plantholt and Tipnis (1991) proved that for any even integer $r$, a regular multigraph $G$ with even order $n$, multiplicity $\mu(G) \leq r$ and degree high relative to $n$ and $r$ is 1-factorable. Here we extend this result to include the case when $r$ is any odd integer. Häggkvist and Perković and Reed (1997) proved that the One-factorization Conjecture for simple graphs is asymptotically true. Our techniques yield an extension of this asymptotic result on simple graphs to a corresponding asymptotic result on multigraphs.

Galvin ([7]) proved that every k-edge-colorable bipartite multigraph is kedge-choosable. Slivnik ([11]) gave a streamlined proof of Galvin's result. A multigraph G is said to be nearly bipartite if it contains a special vertex Vs such that G Vs is a bipartite multigraph. We use the technique in Slivnik's proof to obtain a list coloring analog of Vizing's theorem ([12]) for nearly bipartite multigraphs, and to obtain an extension (suggested by Woodall ([13])) of Galvin's result to multigraphs whose underlying simple graph is bipartite 'plus one edge'. We also prove that for any nearly bipartite multigraph G with special vertex Vs of degree at most six, if G is k-edge-colorable then G is k-edge-choosable.

For a positive integer d, the usual d-dimensional cube Q(d) is defined to be the graph (K-2)(d), the Cartesian product of d copies of K-2. We define the generalized cube Q(d,k) to be the graph (K-k)(d) for positive integers d and k. We investigate the decompositions of the complete graph K-kd and the complete k-partite graph Kkxkd-1 into generalized cubes when k is the power of a prime and d is any positive integer, and some generalizations. We also use these results to show that Q(5) divides K-96.

The maximum of the maximum degree and the “odd set quotients” provides a well-known lower bound φ(G) for the chromatic index of a multigraph G. Plantholt proved that if G is a multigraph of order at most 8, its chromatic index equals φ(G) and that if G is a multigraph of order 10, the chromatic index of G cannot exceed φ(G) + 1. We identify those multigraphs G of order 9 and 10 whose chromatic index equals φ(G) + 1, thus completing the determination of the chromatic index of all multigraphs of order at most 10.