Regular Subgraphs Research Articles

Graph editing problems with extended regularity constraints

Graph editing problems offer an interesting perspective on sub- and supergraph identification problems for a large variety of target properties. They have also attracted significant attention in recent years, particularly in the area of parameterized complexity as the problems have rich parameter ecologies.In this paper we examine generalisations of the notion of editing a graph to obtain a regular subgraph. In particular we extend the notion of regularity to include two variants of edge-regularity along with the unifying constraint of strong regularity. We present a number of results, with the central observation that these problems retain the general complexity profile of their regularity-based inspiration: when the number of edits k and the maximum degree r are taken together as a combined parameter, the problems are tractable (i.e. in FPT), but are otherwise intractable.We also examine variants of the basic editing to obtain a regular subgraph problem from the perspective of parameterizing by the treewidth of the input graph. In this case the treewidth of the input graph essentially becomes a limiting parameter on the natural k+r parameterization.

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.

Counting and packing Hamilton cycles in dense graphs and oriented graphs

We present a general method for counting and packing Hamilton cycles in dense graphs and oriented graphs, based on permanent estimates. We utilize this approach to prove several extremal results. In particular, we show that every nearly cn-regular oriented graph on n vertices with c>3/8 contains (cn/e)n(1+o(1))n directed Hamilton cycles. This is an extension of a result of Cuckler, who settled an old conjecture of Thomassen about the number of Hamilton cycles in regular tournaments. We also prove that every graph G on n vertices of minimum degree at least (1/2+o(1))n contains at least (1−o(1))regeven(G)/2 edge-disjoint Hamilton cycles, where regeven(G) is the maximum even degree of a spanning regular subgraph of G. This establishes an approximate version of a conjecture of Kühn, Lapinskas and Osthus.

Induced Cycles in Graphs

The maximum number vertices of a graph G inducing a 2-regular subgraph of G is denoted by $$c_\mathrm{ind}(G)$$cind(G). We prove that if G is an r-regular graph of order n, then $$c_\mathrm{ind}(G) \ge \frac{n}{2(r-1)} + \frac{1}{(r-1)(r-2)}$$cind(G)źn2(r-1)+1(r-1)(r-2) and we prove that if G is a cubic, claw-free graph on order n, then $$c_\mathrm{ind}(G) > \frac{13}{20}n$$cind(G)>1320n and this bound is asymptotically best possible.

Order of the largest Sachs subgraphs in graphs

Let G be a graph. A subgraph H of G is called a Sachs subgraph if each component of H is either a copy of or a 2-regular subgraph of G. The order of the largest Sachs subgraph of G is called the perrank of G. A graph G of order n has full perrank if In this article, we characterize the family of all graphs of order n whose permanents of their adjacency matrices are 1. Then we prove that the line graph of G, L(G), has full perrank, unless G is isomorphic to some special trees.

Decompositions of Complete Graphs into Bipartite 2-Regular Subgraphs

It is shown that if $G$ is any bipartite 2-regular graph of order at most $n/2$ or at least $n-2$, then the obvious necessary conditions are sufficient for the existence of a decomposition of the complete graph of order $n$ into a perfect matching and edge-disjoint copies of $G$.

Regular subgraphs of uniform hypergraphs

We prove that for every integer r≥2, an n-vertex k-uniform hypergraph H containing no r-regular subgraphs has at most (1+o(1))(n−1k−1) edges if k≥r+1 and n is sufficiently large. Moreover, if r∈{3,4}, r|k and k, n are both sufficiently large, then the maximum number of edges in an n-vertex k-uniform hypergraph containing no r-regular subgraphs is exactly (n−1k−1), with equality only if all edges contain a specific vertex v. We also ask some related questions.

Induced 2-regular subgraphs in [formula omitted]-chordal cubic graphs

We show that a cubic graph G of order n has an induced 2-regular subgraph of order at least •n−24−4k, if G has no induced cycle of length more than k,•5n+68, if G has no induced cycle of length more than 4, and n>6, and•(14+ϵ)n, if the independence number of G is at most (38−ϵ)n. To show the second result we give a precise structural description of cubic 4-chordal graphs.

Graph Mining Using gSpan: Graph-Based Substructure Pattern Mining

We have explored new methodologies for regular graph-based pattern exploration in graph datasets and studied a novel algorithm called gSpan (graph-based substructure pattern mining), which finds frequent substructures without candidate production. gSpan fabricates another lexicographic arrangement between diagrams and maps every chart to a kind smaller depth-first search (DFS) code as its standard label. Taking into account this lexico- realistic request, gSpan embraces the depth-First search approach to mine regular associated subgraphs efficiently. Our performance study demonstrates that gSpan significantly beats previous calculations, once in a while by a request of scale.

Largest independent sets of certain regular subgraphs of the derangement graph

The derangement graph is the Cayley graph on the symmetric group $$\mathcal {S}_{n}$$Sn whose generating set $$D_{n}$$Dn is the set of permutations on $$[n]=\{1, \ldots , n\}$$[n]={1,?,n} without any 1-cycle. For any fixed positive integer $$k \le n$$k≤n, the Cayley graph generated by the subset of $$D_{n}$$Dn consisting of permutations without any i-cycles for all $$1 \le i \le k$$1≤i≤k is a regular subgraph of the derangement graph. In this paper, we determine the smallest eigenvalue of these subgraphs and show that the set of all the largest independent sets in these subgraphs is equal to the set of all the largest independent sets in the derangement graph, provided n is sufficiently large in terms of k.

Edge-Colorings of 4-Regular Graphs with the Minimum Number of Palettes

A proper edge-coloring of a graph G is an assignment of colors to the edges of G such that adjacent edges receive distinct colors. A proper edge-coloring defines at each vertex the set of colors of its incident edges. Following the terminology introduced by Hor?ak, Kalinowski, Meszka and Wo?niak, we call such a set of colors the palette of the vertex. What is the minimum number of distinct palettes taken over all proper edge-colorings of G? A complete answer is known for complete graphs and cubic graphs. We study in some detail the problem for 4-regular graphs. In particular, we show that certain values of the palette index imply the existence of an even cycle decomposition of size 3 (a partition of the edge-set of a graph into 3 2-regular subgraphs whose connected components are cycles of even length). This result can be extended to 4d-regular graphs. Moreover, in studying the palette index of a 4-regular graph, the following problem arises: does there exist a 4-regular graph whose even cycle decompositions cannot have size smaller than 4?

On 3-Regular Bipancyclic Subgraphs of Hypercubes

The n-dimensional hypercube Qn is bipancyclic; that is, it contains a cycle of every even length from 4 to 2n. In this paper, we prove that Qn (n≥3) contains a 3-regular, 3-connected, bipancyclic subgraph with l vertices for every even l from 8 to 2n except 10.

Extending a perfect matching to a Hamiltonian cycle

Graph Theory Ruskey and Savage conjectured that in the d-dimensional hypercube, every matching M can be extended to a Hamiltonian cycle. Fink verified this for every perfect matching M, remarkably even if M contains external edges. We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d ≥7 and every k, where 7 ≤k ≤d, the d-dimensional hypercube contains a k-regular spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle. We do not know if this result can be extended to k=4,5,6. It cannot be extended to k=3. Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Also, we do not know if there are graphs of girth at least 5 with this matching-extendability property.

Decompositions of graphs into trees, forests, and regular subgraphs

A conjecture by A. Hoffmann-Ostenhof suggests that any connected cubic graph G contains a spanning tree T for which every component of G−E(T) is a K1, a K2, or a cycle. We show that any cubic graph G contains a spanning forest F for which every component of G−E(F) is a K2 or a cycle, and that any connected graph G≠K1 with maximal degree at most 3 contains a spanning forest F without isolated vertices for which every component of G−E(F) is a K1, a K2 or a cycle. We also prove a related statement about path-factorizations of graphs with maximal degree 3.

On graphs with in strongly regular local subgraphs having parameters (88, 27, 6, 9)

On graphs with in strongly regular local subgraphs having parameters (88, 27, 6, 9)

3-Regular subgraphs and (3,1)-colorings of 4-regular pseudographs

Let G be a 4-regular pseudograph. Refer as a (3, 1)-coloring of G to a coloring of its edges by several colors such that three edges of one color and one of another meet at each vertex. The properties of (3, 1)-colorings are closely connected with the presence of 3-regular subgraphs in the graph. We prove that each connected 4-regular pseudograph containing a 3-regular subgraph admits some (3, 1)-coloring. Moreover, each 4-regular pseudograph without multiple edges (but possibly with loops) has (3, 1)-coloring, which indirectly confirms the (unproven) conjecture that every such graph contains a 3-regular subgraph. We also analyze the question of the least number of colors for a (3, 1)-coloring of a given 4-regular graph. In conclusion, we prove that the existence of a (3, 1)-coloring satisfying additional requirements (an ordered (3, 1)-coloring) is equivalent to the existence of a 3-regular subgraph.

A recursive construction of the regular exceptional graphs with least eigenvalue –2

In spectral graph theory a graph with least eigenvalue -2 is exceptional if it is connected, has least eigenvalue greater than or equal to -2 , and it is not a generalized line graph. A (\kappa,\tau) -regular set S of a graph is a vertex subset, inducing a \kappa -regular subgraph such that every vertex not in S has \tau neighbors in S . We present a recursive construction of all regular exceptional graphs as successive extensions by regular sets.

On graphs with strongly regular local subgraphs with parameters (111, 30, 5, 9) or (169, 42, 5, 12)

On graphs with strongly regular local subgraphs with parameters (111, 30, 5, 9) or (169, 42, 5, 12)

On graphs with strongly regular local subgraphs with parameters (196, 45, 4, 12)

On graphs with strongly regular local subgraphs with parameters (196, 45, 4, 12)

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.