Subgraph Of Degree Research Articles

How does the core sit inside the mantle?

The k-core, defined as the largest subgraph of minimum degree k, of the random graph G(n,p) has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [Journal of Combinatorial Theory, Series B 67 (1996) 111–151] determined the threshold dk for the appearance of an extensive k-core. Here we derive a multi-type Galton-Watson branching process that describes precisely how the k-core is “embedded” into the random graph for any k≥3 and any fixed average degree d=np>dk. This generalises prior results on, e.g., the internal structure of the k-core.

Monochromatic bounded degree subgraph partitions

Let F={F1,F2,…} be a sequence of graphs such that Fn is a graph on n vertices with maximum degree at most Δ. We show that there exists an absolute constant C such that the vertices of any 2-edge-colored complete graph can be partitioned into at most 2CΔlogΔ vertex disjoint monochromatic copies of graphs from F. If each Fn is bipartite, then we can improve this bound to 2CΔ; this result is optimal up to the constant C.

A Precise Threshold for Quasi-Ramsey Numbers

We consider the variation of Ramsey numbers introduced by Erdös and Pach [J. Graph Theory, 7 (1983), pp. 137--147], where instead of seeking complete or independent sets we only seek a $t$-homogeneous set, a vertex subset that induces a subgraph of minimum degree at least $t$ or the complement of such a graph. For any $\nu > 0$ and positive integer $k$, we show that any graph $G$ or its complement contains as an induced subgraph some graph $H$ on $\ell \ge k$ vertices with minimum degree at least $\frac12(\ell-1) + \nu$ provided that $G$ has at least $k^{\Omega(\nu^2)}$ vertices. We also show this to be the best possible in a sense. This may be viewed as correction to a result claimed in [P. Erdös and J. Pach, J. Graph Theory, 7 (1983), pp. 137--147]. For the above result, we permit $H$ to have order at least $k$. In the harder problem, where we insist that $H$ have exactly $k$ vertices, we do not obtain sharp results, although we show a way to translate results of one form of the problem to the other.

Optimal Local Routing on Delaunay Triangulations Defined by Empty Equilateral Triangles

We present a deterministic local routing algorithm that is guaranteed to find a path between any pair of vertices in a half-$\theta_6$-graph (the half-$\theta_6$-graph is equivalent to the Delaunay triangulation where the empty region is an equilateral triangle). The length of the path is at most $5/\sqrt{3} \approx 2.887$ times the Euclidean distance between the pair of vertices. Moreover, we show that no local routing algorithm can achieve a better routing ratio, thereby proving that our routing algorithm is optimal. This is somewhat surprising because the spanning ratio of the half-$\theta_6$-graph is 2, meaning that even though there always exists a path whose length is at most twice the Euclidean distance, we cannot always find such a path when routing locally. Since every triangulation can be embedded in the plane as a half-$\theta_6$-graph using $O(\log n)$ bits per vertex coordinate via Schnyder's embedding scheme [W. Schnyder, Embedding planar graphs on the grid, in Proceedings of the 1st Annual ACM--SIAM Symposium on Discrete Algorithms (SODA 1990), ACM, New York, SIAM, Philadelphia, 1990, pp. 138--148], our result provides a competitive local routing algorithm for every such embedded triangulation. Finally, we show how our routing algorithm can be adapted to provide a routing ratio of $15/\sqrt{3} \approx 8.660$ on two bounded degree subgraphs of the half-$\theta_6$-graph.

Polarity graphs revisited

Polarity graphs, also known as Brown graphs, and their minor modifications are the largest currently known graphs of diameter 2 and a given maximum degree d such that d − 1 is a prime power larger than 5 . In view of the recent interest in the degree-diameter problem restricted to vertex-transitive and Cayley graphs we investigate ways of turning the (non-regular) polarity graphs to large vertex-transitive graphs of diameter 2 and given degree. We review certain properties of polarity graphs, giving new and shorter proofs. Then we show that polarity graphs of maximum even degree d cannot be spanning subgraphs of vertex-transitive graphs of degree at most d + 2 . If d − 1 is a power of 2 , there are two large vertex-transitive induced subgraphs of the corresponding polarity graph, one of degree d − 1 and the other of degree d − 2 . We show that the subgraphs of degree d − 1 cannot be extended to vertex-transitive graphs of diameter 2 by adding a relatively small non-edge orbital. On the positive side, we prove that the subgraphs of degree d − 2 can be extended to the largest currently known Cayley graphs of given degree and diameter 2 found by Šiagiová and the second author [ J. Combin. Theory Ser. B 102 (2012), 470–473].

Large Feedback Arc Sets, High Minimum Degree Subgraphs, and Long Cycles in Eulerian Digraphs

A minimum feedback arc set of a directed graph G is a smallest set of arcs whose removal makes G acyclic. Its cardinality is denoted by β(G). We show that a simple Eulerian digraph with n vertices and m arcs has β(G) ≥ m2/2n2+m/2n, and this bound is optimal for infinitely many m, n. Using this result we prove that a simple Eulerian digraph contains a cycle of length at most 6n2/m, and has an Eulerian subgraph with minimum degree at least m2/24n3. Both estimates are tight up to a constant factor. Finally, motivated by a conjecture of Bollobás and Scott, we also show how to find long cycles in Eulerian digraphs.

Random walks which prefer unvisited edges: Exploring high girth even degree expanders in linear time

AbstractLet be a connected graph with vertices. A simple random walk on the vertex set of G is a process, which at each step moves from its current vertex position to a neighbouring vertex chosen uniformly at random.We consider a modified walk which, whenever possible, chooses an unvisited edge for the next transition; and makes a simple random walk otherwise. We call such a walk an edge‐process (or E‐process). The rule used to choose among unvisited edges at any step has no effect on our analysis. One possible method is to choose an unvisited edge uniformly at random, but we impose no such restriction.For the class of connected even degree graphs of constant maximum degree, we bound the vertex cover time of the E‐process in terms of the edge expansion rate of the graph G, as measured by eigenvalue gap of the transition matrix of a simple random walk on G.A vertex v is ℓ‐good, if any even degree subgraph containing all edges incident with v contains at least ℓ vertices. A graph G is ℓ‐good, if every vertex has the ℓ‐good property. Let G be an even degree ℓ‐good expander of bounded maximum degree. Any E‐process on G has vertex cover time urn:x-wiley:10429832:media:rsa20504:rsa20504-math-0004 This is to be compared with the lower bound on the cover time of any connected graph by a weighted random walk. Our result is independent of the rule used to select the order of the unvisited edges, which could, for example, be chosen on‐line by an adversary.As no walk based process can cover an n vertex graph in less than n – 1 steps, the cover time of the E‐process is of optimal order when . With high probability random r‐regular graphs, even, have . Thus the vertex cover time of the E‐process on such graphs is . © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 36–54, 2015

On a greedy 2‐matching algorithm and Hamilton cycles in random graphs with minimum degree at least three

ABSTRACTWe describe and analyse a simple greedy algorithm 2greedy that finds a good 2‐matching M in the random graph when . A 2‐matching is a spanning subgraph of maximum degree two and G is drawn uniformly from graphs with vertex set , cn edges and minimum degree at least three. By good we mean that M has components. We then use this 2‐matching to build a Hamilton cycle in time w.h.p. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 443‐497, 2014

On the threshold for k-regular subgraphs of random graphs

The k-core of a graph is the largest subgraph of minimum degree at least k. We show that for k sufficiently large, the threshold for the appearance of a k-regular subgraph in the Erdős-Renyi random graph model G(n,p) is at most the threshold for the appearance of a nonempty (k+2)-core. In particular, this pins down the point of appearance of a k-regular subgraph to a window for p of width roughly 2/n for large n and moderately large k. The result is proved by using Tutte’s necessary and sufficient condition for a graph to have a k-factor.

Vertex decompositions of sparse graphs into an independent vertex set and a subgraph of maximum degree at most 1

A graph G is (1, 0)-colorable if its vertex set can be partitioned into subsets V1 and V0 so that in G[V1] every vertex has degree at most 1, while G[V0] is edgeless. We prove that every graph with maximum average degree at most \(\tfrac{{12}} {5} \) is (1, 0)-colorable. In particular, every planar graph with girth at least 12 is (1, 0)-colorable. On the other hand, we construct graphs with the maximum average degree arbitrarily close (from above) to \(\tfrac{{12}} {5} \) which are not (1, 0)-colorable.

Parameterized complexity of finding small degree-constrained subgraphs

In this article we study the parameterized complexity of problems consisting in finding degree-constrained subgraphs, taking as the parameter the number of vertices of the desired subgraph. Namely, given two positive integers d and k, we study the problem of finding a d-regular (induced or not) subgraph with at most k vertices and the problem of finding a subgraph with at most k vertices and of minimum degree at least d. The latter problem is a natural parameterization of the d-girth of a graph (the minimum order of an induced subgraph of minimum degree at least d). We first show that both problems are fixed-parameter intractable in general graphs. More precisely, we prove that the first problem is W [ 1 ] -hard using a reduction from Multi-Color Clique. The hardness of the second problem (for the non-induced case) follows from an easy extension of an already known result. We then provide explicit fixed-parameter tractable (FPT) algorithms to solve these problems in graphs with bounded local treewidth and graphs with excluded minors, using a dynamic programming approach. Although these problems can be easily defined in first-order logic, hence by the results of Frick and Grohe (2001) [23] are FPT in graphs with bounded local treewidth and graphs with excluded minors, the dependence on k of our algorithms is considerably better than the one following from Frick and Grohe (2001) [23].

A Lebesgue's type theorem on toroidal graphs and its application to linear coloring

A coloring of a graph <italic>G</italic> is said to be linear if any two color classes induce an acyclic subgraph of maximum degree at most 2. In this paper, we first prove a Lebesgue's type theorem concerning the structure of toroidal graphs. As a consequence of this result, we show that every toroidal graph <italic>G</italic> of girth at least 5 can be 「(△(<italic>G</italic>))/2」+ 4)-linear list colorable, unless △(<italic>G</italic>) = 4 and G has a subgraph containing only faces of degree five of which each is incident with three vertices of degree 3 and two vertices of degree 4. This generalizes and improves some known results.

Trees in greedy colorings of hypergraphs

It is a well-known proposition that every graph of chromatic number larger than t contains every tree with t edges. The ‘standard’ reasoning is that such a graph must contain a subgraph of minimum degree at least t . Bohman, Frieze, and Mubayi noticed that, although this argument does not work for hypergraphs, it is still possible that the proposition holds for hypergraphs as well. Indeed, Loh recently proved that every uniform hypergraph of chromatic number larger than t contains every hypertree with t edges. Here we observe that the basic property of the well-known greedy algorithm immediately implies a much more general result (with a conceptually simpler proof): if the greedy algorithm colors the vertices of an r -uniform hypergraph with more than t colors then the hypergraph contains every r -uniform hypertree with t edges.

Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k

AbstractA graph G is (k,0)‐colorable if its vertices can be partitioned into subsets V1 and V2 such that in G[V1] every vertex has degree at most k, while G[V2] is edgeless. For every integer k⩾0, we prove that every graph with the maximum average degree smaller than (3k+4)/(k+2) is (k,0)‐colorable. In particular, it follows that every planar graph with girth at least 7 is (8, 0)‐colorable. On the other hand, we construct planar graphs with girth 6 that are not (k,0)‐colorable for arbitrarily large k. © 2009 Wiley Periodicals, Inc. J Graph Theory 65:83–93, 2010

The t-Improper Chromatic Number of Random Graphs

We consider the t-improper chromatic number of the Erdős–Rényi random graph Gn,p. The t-improper chromatic number χt(G) is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most t. If t = 0, then this is the usual notion of proper colouring. When the edge probability p is constant, we provide a detailed description of the asymptotic behaviour of χt(Gn,p) over the range of choices for the growth of t = t(n).

A general critical condition for the emergence of a giant component in random graphs with given degrees

In this contribution, we investigate the giant component problem in random graphs with a given degree sequence. We generalize the critical condition of Molloy and Reed [Molloy, M., and B. Reed, A critical point for random graphs with given degree sequence, Random Structures Algorithms 6 (1995), 161-179], which determines the existence of a giant component in such a random graph, in order to include degree sequences with heavy tails. We show that the quantity which determines the existence of a giant component is the value of the smallest fixed point inside the interval [0, 1] of the generating function F(s)=∑i⩾1δisi−1, where δi is the asymptotic proportion of the total degree contained in vertices of degree i. Moreover, we show that this quantity also determines the existence of a core (i.e., the maximal subgraph of minimum degree at least 2) that has linear total degree.

Improper colouring of (random) unit disk graphs

For any graph G, the k-improper chromatic number χ k ( G ) is the smallest number of colours used in a colouring of G such that each colour class induces a subgraph of maximum degree k. We investigate χ k for unit disk graphs and random unit disk graphs to generalise results of McDiarmid and Reed [Colouring proximity graphs in the plane, Discrete Math. 199(1–3) (1999) 123–137], McDiarmid [Random channel assignment in the plane, Random Structures Algorithms 22(2) (2003) 187–212], and McDiarmid and Müller [On the chromatic number of random geometric graphs, submitted for publication].

The t-improper chromatic number of random graphs

We consider the t-improper chromatic number of the Erdős–Renyi random graph Gn,p. The t-improper chromatic number χt(G) is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most t. If t = 0, then this is the usual notion of proper colouring. When the edge probability p is constant, we provide a detailed description of the asymptotic behaviour of χt(Gn,p) over the range of choices for the growth of t = t(n).

On Maximum Cost $K_{t,t}$‐Free t‐Matchings of Bipartite Graphs

Frank examined the maximum $K_{t,t}$-free $t$-matching problem of simple bipartite graphs. As the $C_6$-free $2$-matching problem is NP-hard (Geelen), this is a promising generalization of restricted $2$-matchings. Given an arbitrary family $\mathcal{T}$ of $K_{t,t}$-subgraphs of the underlying graph, a $\mathcal{T}$-free $t$-matching is a subgraph of maximum degree at most $t$ that contains no member of $\mathcal{T}$. We show that the maximum size $\mathcal{T}$-free $t$-matching problem also admits a nice min-max formula. Given an integer cost function on the edge-set which is vertex-induced on any member of $\mathcal{T}$, we also show an integer min-max formula for the maximum cost of $\mathcal{T}$-free $t$-matchings. As the maximum cost $C_4$-free 2-matching problem is NP-hard (Kira´ly), we cannot expect a nice characterization in general.

Infinite geodesic paths and fibers, new topological invariants in periodic graphs

Rings are well known invariants of nets. In this work, a generalization of the concepts of cycles and rings is introduced. Infinite paths in periodic graphs are defined as connected, acyclic, regular subgraphs of degree two; geodesics are defined as infinite paths such that the unique path between any pair of vertices is a geodesic path in the whole graph. An infinite path can be thought of as an infinite cycle and a geodesic as an infinite ring. In a further step, a geodesic fiber is defined as a minimal 1-periodic subgraph that contains all geodesic paths between any pair of its vertices. Geodesic fibers are topological invariants of periodic graphs whose labeled quotient graphs are subgraphs of the labeled quotient graph of the whole graph; the paper describes applications of geodesic fibers to the analysis of the automorphisms of minimal nets, crystallographic and non-crystallographic nets.