Subgraph Of Degree Research Articles

A Note on Internal Partitions: The 5-Regular Case and Beyond

An internal or friendly partition of a graph is a partition of the vertex set into two nonempty sets so that every vertex has at least as many neighbours in its own class as in the other one. It has been shown that apart from finitely many counterexamples, every 3, 4 or 6-regular graph has an internal partition. In this note we focus on the 5-regular case and show that among the subgraphs of minimum degree at least 3 of 5-regular graphs, there are some which have small intersection. We also discuss the existence of internal partitions in some families of Cayley graphs, notably we determine all 5-regular Abelian Cayley graphs which do not have an internal partition.

On the number of perfect matchings in random polygonal chains

Abstract Let G G be a graph. A perfect matching of G G is a regular spanning subgraph of degree one. Enumeration of perfect matchings of a (molecule) graph is interest in chemistry, physics, and mathematics. But the enumeration problem of perfect matchings for general graphs (even in bipartite graphs) is non-deterministic polynomial (NP)-hard. Xiao et al. [C. Xiao, H. Chen, L. Liu, Perfect matchings in random pentagonal chains, J. Math. Chem. 55 (2017), 1878–1886] have studied the problem of perfect matchings for random odd-polygonal chain (i.e., with odd polygons). In this article, we further present simple counting formulae for the expected value of the number of perfect matchings in random even-polygonal chains (i.e., with even polygons). Based on these formulae, we obtain the average values of the number for perfect matchings with respect to the set of all even-polygonal chains with n n polygons.

Relating dissociation, independence, and matchings

A dissociation set in a graph is a set of vertices inducing a subgraph of maximum degree at most 1. Computing the dissociation number diss ( G ) of a given graph G , defined as the order of a maximum dissociation set in G , is algorithmically hard even when G is restricted to be bipartite. Recently, Hosseinian and Butenko proposed a simple 4 3 -approximation algorithm for the dissociation number problem in bipartite graphs. Their result relies on the inequality diss ( G ) ≤ 4 3 α ( G − M ) , where G is a bipartite graph, M is a maximum matching in G , and α ( G − M ) denotes the independence number of G − M . We show that the pairs ( G , M ) for which this inequality holds with equality can be recognized efficiently, and that a maximum dissociation set can be determined for them efficiently. The dissociation number of a graph G satisfies max { α ( G ) , 2 ν s ( G ) } ≤ diss ( G ) ≤ α ( G ) + ν s ( G ) ≤ 2 α ( G ) , where ν s ( G ) denotes the induced matching number of G . We show that deciding whether diss ( G ) equals any of the four terms α ( G ) , 2 ν s ( G ) , α ( G ) + ν s ( G ) , and 2 α ( G ) is NP-hard.

Vizing's and Shannon's Theorems for Defective Edge Colouring

We call a multigraph $(k,d)$-edge colourable if its edge set can be partitioned into $k$ subgraphs of maximum degree at most $d$ and denote as $\chi'_{d}(G)$ the minimum $k$ such that $G$ is $(k,d)$-edge colourable. We prove that for every odd integer $d$, every multigraph $G$ with maximum degree $\Delta$ is $(\lceil \frac{3\Delta - 1}{3d - 1} \rceil, d)$-edge colourable and this bound is attained for all values of $\Delta$ and $d$. An easy consequence of Vizing's Theorem is that, for every (simple) graph $G,$ $\chi'_{d}(G) \in \{ \lceil \frac{\Delta}{d} \rceil, \lceil \frac{\Delta+1}{d} \rceil \}$. We characterize the values of $d$ and $\Delta$ for which it is NP-complete to compute $\chi'_d(G)$. These results generalize classic results on the chromatic index of a graph by Shannon, Holyer, Leven and Galil and extend a result of Amini, Esperet and van den Heuvel.

Component Games on Random Graphs

In the (1:b) component game played on a graph G, two players, Maker and Breaker, alternately claim 1 and b previously unclaimed edges of G, respectively. Maker’s aim is to maximise the size of a largest connected component in her graph, while Breaker is trying to minimise it. We show that the outcome of the game on the binomial random graph is strongly correlated with the appearance of a nonempty (b +2)-core in the graph. For any integer k, the k-core of a graph is its largest subgraph of minimum degree at least k. Pittel, Spencer and Wormald showed in 1996 that for any k ≥ 3 there exists a constant ck, which they determine, such that p = ck/n is the threshold function for the appearance of the k-core in $$G \sim {\cal G}\left( {n,p} \right)$$ . More precisely, $$G \sim {\cal G}\left( {n,c/n} \right)$$ has WHP a linear-size k-core for constant c>ck, and an empty k-core when c<ck. We show that for any positive constant integer b, when playing the (1:b) component game on $$G \sim {\cal G}\left( {n,c/n} \right)$$ , Maker can WHP build a linear-size component if c > cb+2, while Breaker can WHP prevent Maker from building larger than polylogarithmic-size components if c< cb+2. For the strategy of Maker when c> cb+2, we utilise known results on the k-core. For Breaker when c< cb+2, we make use of a result of Achlioptas and Molloy (sketching its proof) that states that after deleting all vertices of degree less than k, and repeating this step a constant number of times, G is WHP shattered into pieces of polylogarithmic size.

On sensitivity in bipartite Cayley graphs

Huang proved that every set of more than half the vertices of the d-dimensional hypercube Qd induces a subgraph of maximum degree at least d, which is tight by a result of Chung, Füredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs.First, we present three infinite families of Cayley graphs of unbounded degree that contain induced subgraphs of maximum degree 1 on more than half the vertices. In particular, this refutes a conjecture of Potechin and Tsang, for which first counterexamples were shown recently by Lehner and Verret. The first family consists of dihedrants and contains a sporadic counterexample encountered earlier by Lehner and Verret. The second family are star graphs, these are edge-transitive Cayley graphs of the symmetric group. All members of the third family are d-regular containing an induced matching on a d2d−1-fraction of the vertices. This is largest possible and answers a question of Lehner and Verret.Second, we consider Huang's lower bound for graphs with subcubes and show that the corresponding lower bound is tight for products of Coxeter groups of type An, I2(2k+1), and most exceptional cases. We believe that Coxeter groups are a suitable generalization of the hypercube with respect to Huang's question.Finally, we show that induced subgraphs on more than half the vertices of Levi graphs of projective planes and of the Ramanujan graphs of Lubotzky, Phillips, and Sarnak have unbounded degree. This gives classes of Cayley graphs with properties similar to the ones provided by Huang's results. However, in contrast to Coxeter groups these graphs have no subcubes.

Minimum k‐cores and the k‐core polytope

AbstractThe minimum k‐core problem asks for the smallest induced subgraph of minimum degree k. It has been shown that this problem is NP‐hard, and thus sophisticated techniques are required to obtain good solutions and approximations. In this article, the minimum k‐core problem is modeled as a binary integer program and relaxed as a linear program. Since the relaxation may yield a non‐integral solution, a branch‐and‐cut framework is used to find an integral optimal solution. It is shown that the edge and cycle transversals of the graph give valid inequalities for the convex hull of the k‐core polytope—which can be further generalized to a family of ‐core transversals. Further, a heuristic for the transversal of the minimal ‐cores is given with its associated valid inequality. Additionally, improved valid inequalities are generated using bounds involving the girth of the graph. Multiple heuristics are explored for finding initial bounds for the branching process utilizing the degree distribution of the graph. Finally, numerical results are given comparing the branch‐and‐bound, branch‐and‐cut, and heuristic techniques.

Trestles in the squares of graphs

We show that the square of every connected S(K1,4)-free graph satisfying a matching condition has a 2-connected spanning subgraph of maximum degree at most 3. Furthermore, we characterise trees whose square has a 2-connected spanning subgraph of maximum degree at most k. This generalises the results on S(K1,3)-free graphs of Henry and Vogler [7] and Harary and Schwenk [6], respectively.

Bounded Degree Spanners of the Hypercube

In this short note we study two questions about the existence of subgraphs of the hypercube $Q_n$ with certain properties. The first question, due to Erdős–Hamburger–Pippert–Weakley, asks whether there exists a bounded degree subgraph of $Q_n$ which has diameter $n$. We answer this question by giving an explicit construction of such a subgraph with maximum degree at most 120.&#x0D; The second problem concerns properties of $k$-additive spanners of the hypercube, that is, subgraphs of $Q_n$ in which the distance between any two vertices is at most $k$ larger than in $Q_n$. Denoting by $\Delta_{k,\infty}(n)$ the minimum possible maximum degree of a $k$-additive spanner of $Q_n$, Arizumi–Hamburger–Kostochka showed that $$\frac{n}{\ln n}e^{-4k}\leq \Delta_{2k,\infty}(n)\leq 20\frac{n}{\ln n}\ln \ln n.$$ We improve their upper bound by showing that $$\Delta_{2k,\infty}(n)\leq 10^{4k} \frac{n}{\ln n}\ln^{(k+1)}n,$$where the last term denotes a $k+1$-fold iterated logarithm.

Bipartite Induced Density in Triangle-Free Graphs

We prove that any triangle-free graph on $n$ vertices with minimum degree at least $d$ contains a bipartite induced subgraph of minimum degree at least $d^2/(2n)$. This is sharp up to a logarithmic factor in $n$. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of $n/d$ and $(2+o(1))\sqrt{n/\log n}$ as $n\to\infty$. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most $O(\min\{\sqrt{n},(n\log n)/d\})$ as $n\to\infty$.&#x0D; Relatedly, we also make two conjectures. First, any triangle-free graph on $n$ vertices has fractional chromatic number at most $(\sqrt{2}+o(1))\sqrt{n/\log n}$ as $n\to\infty$. Second, any triangle-free graph on $n$ vertices has list chromatic number at most $O(\sqrt{n/\log n})$ as $n\to\infty$.

Dense Induced Bipartite Subgraphs in Triangle-Free Graphs

The problem of finding dense induced bipartite subgraphs in $H$-free graphs has a long history, and was posed 30 years ago by Erd\H{o}s, Faudree, Pach and Spencer. In this paper, we obtain several results in this direction. First we prove that any $K_t$-free graph with minimum degree at least $d$ contains an induced bipartite subgraph of minimum degree at least $c_t \log d/\log \log d$, confirming (asymptotically) several conjectures by Esperet, Kang and Thomass\'e. Complementing this result, we further obtain optimal bounds for this problem in the case of dense triangle-free graphs, and we also answer a question of Erd\H{o}s, Janson, {\L}uczak and Spencer.

Separation Choosability and Dense Bipartite Induced Subgraphs

AbstractWe study a restricted form of list colouring, for which every pair of lists that correspond to adjacent vertices may not share more than one colour. The optimal list size such that a proper list colouring is always possible given this restriction, we call separation choosability. We show for bipartite graphs that separation choosability increases with (the logarithm of) the minimum degree. This strengthens results of Molloy and Thron and, partially, of Alon. One attempt to drop the bipartiteness assumption precipitates a natural class of Ramsey-type questions, of independent interest. For example, does every triangle-free graph of minimum degree d contain a bipartite induced subgraph of minimum degree Ω(log d) as d→∞?

A proof of a conjecture of Erdős, Faudree, Rousseau and Schelp on subgraphs of minimum degree k

Erdős, Faudree, Rousseau and Schelp observed the following fact for every fixed integer k≥2: Every graph on n≥k−1 vertices with at least (k−1)(n−k+2)+(k−22) edges contains a subgraph with minimum degree at least k. However, there are examples in which the whole graph is the only such subgraph. Erdős et al. conjectured that having just one more edge implies the existence of a subgraph on at most (1−εk)n vertices with minimum degree at least k, where εk>0 depends only on k. We prove this conjecture, using and extending ideas of Mousset, Noever and Škorić.

Smaller Subgraphs of Minimum Degree $k$

In 1990 Erdős, Faudree, Rousseau and Schelp proved that for $k \ge 2$, every graph with $n \ge k+1$ vertices and $(k-1)(n-k+2)+\binom{k-2}{2}+1$ edges contains a subgraph of minimum degree $k$ on at most $n-\sqrt{n/6k^3}$ vertices. They conjectured that it is possible to remove at least $\epsilon_k n$ many vertices and remain with a subgraph of minimum degree $k$, for some $\epsilon_k&gt;0$. We make progress towards their conjecture by showing that one can remove at least order of $\Omega(n/\log n)$ many vertices.

Embedding spanning bounded degree subgraphs in randomly perturbed graphs

We study the model of randomly perturbed dense graphs, which is the union of any graph Gα with minimum degree αn and the binomial random graph G(n,p). For p=ω(n−2/(Δ+1)), we show that Gα∪G(n,p) contains any single spanning graph with maximum degree Δ. As in previous results concerning this model, the bound for p we use is lower by a log-term in comparison to the bound known to be needed to find the same subgraph in G(n,p) alone.

On partitions of graphs under degree constraints

Let s,t be two integers, and let g(s,t) denote the minimum integer such that the vertex set of a graph of minimum degree at least g(s,t) can be partitioned into two nonempty sets which induce subgraphs of minimum degree at least s and t, respectively. In this paper, it is shown that, (1) for positive integers s and t, g(s,t)≤s+t on (K4−e)-free graphs except K3, and (2) for integers s≥2 and t≥2, g(s,t)≤s+t−1 on triangle-free graphs in which no two quadrilaterals share edges. Our first conclusion generalizes a result of Kaneko (1998), and the second generalizes a result of Diwan (2000).

How does the core sit inside the mantle?

AbstractThe k‐core, defined as the maximal subgraph of minimum degree at least k, of the random graph has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [J Combin Theory Ser B 67 (1996), 111–151] determined the threshold dk for the appearance of an extensive k‐core.The aim of the present paper is to describe how the k‐core is “embedded” into the random graph in the following sense. Let and fix . Colour each vertex that belongs to the k‐core of in black and all remaining vertices in white. Here we derive a multi‐type branching process that describes the local structure of this coloured random object as n tends to infinity. This generalises prior results on, e.g., the internal structure of the k‐core.In the physics literature it was suggested to characterize the core by means of a message passing algorithm called Warning Propagation. Ibrahimi, Kanoria, Kraning and Montanari [Ann Appl Probab 25 (2015), 2743–2808] used this characterization to describe the 2‐core of random hypergraphs. To derive our main result we use a similar approach. A key observation is that a bounded number of iterations of this algorithm is enough to give a good approximation of the k‐core. Based on this the study of the k‐core reduces to the analysis of Warning Propagation on a suitable Galton‐Watson tree. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 459–482, 2017

Towards a Version of Ohba’s Conjecture for Improper Colorings

The d-improper chromatic number $$\chi ^d(G)$$?d(G) of a graph G is the minimum number of colors to color G such that each color class induces a subgraph of maximum degree at most d. The d-improper choice number is the list-coloring version of this concept. A graph is called d-improperly chromatic-choosable if its d-improper choice number equals its d-improper chromatic number. As a generalization of a recently confirmed conjecture of Ohba that every graph G with $$|V(G)| \le 2\chi (G)+1$$|V(G)|≤2?(G)+1 is chromatic-choosable, Yan et al. proposed an improper coloring-based version of Ohba's conjecture: every graph G with $$|V(G)|\le (d+2)\chi ^d(G)+(d+1)$$|V(G)|≤(d+2)?d(G)+(d+1) is d-improperly chromatic-choosable. In this paper, using graph theoretic and probabilistic methods we prove that the conjecture is true for $$|V(G)| \le (d+\frac{3}{2})\chi ^d(G)+\frac{d}{2}$$|V(G)|≤(d+32)?d(G)+d2. We also construct a family of graphs G with $$|V(G)|=(d+3)\chi ^d(G)+(d+3)$$|V(G)|=(d+3)?d(G)+(d+3) which are not d-improperly chromatic-choosable.

Graphs without proper subgraphs of minimum degree 3 and short cycles

We study graphs on n vertices which have 2nź2 edges and no proper induced subgraphs of minimum degree 3. Erdźs, Faudree, Gyarfas, and Schelp conjectured that such graphs always have cycles of lengths 3,4,5,...,C(n) for some function C(n) tending to in finity. We disprove this conjecture, resolve a related problem about leaf-to-leaf path lengths in trees, and characterize graphs with n vertices and 2nź2 edges, containing no proper subgraph of minimum degree 3.

A novel topological centrality measure capturing biologically important proteins.

Topological centrality in protein interaction networks and its biological implications have widely been investigated in the past. In the present study, a novel metric of centrality-weighted sum of loads eigenvector centrality (WSL-EC)-based on graph spectra is defined and its performance in identifying topologically and biologically important nodes is comparatively investigated with common metrics of centrality in a human protein-protein interaction network. The metric can capture nodes from peripherals of the network differently from conventional eigenvector centrality. Different metrics were found to selectively identify hub sets that are significantly associated with different biological processes. The widely accepted metrics degree centrality, betweenness centrality, subgraph centrality and eigenvector centrality are subject to a bias towards super-hubs, whereas WSL-EC is not affected by the presence of super-hubs. WSL-EC outperforms other metrics of centrality in detecting biologically central nodes such as pathogen-interacting, cancer, ageing, HIV-1 or disease-related proteins and proteins involved in immune system processes and autoimmune diseases in the human interactome.