A note on balanced bipartitions

Upper bounds on minimum balanced bipartitions

We consider a variation of the supermarket model in which the servers can communicate with their neighbors and where the neighborhood relationships are described in terms of a suitable graph. Tasks with unit-exponential service time distributions arrive at each vertex as independent Poisson processes with rate $\lambda$, and each task is irrevocably assigned to the shortest queue among the one it first appears and its $d-1$ randomly selected neighbors. This model has been extensively studied when the underlying graph is a clique in which case it reduces to the well known power-of-$d$ scheme. In particular, results of Mitzenmacher (1996) and Vvedenskaya et al. (1996) show that as the size of the clique gets large, the occupancy process associated with the queue-lengths at the various servers converges to a deterministic limit described by an infinite system of ordinary differential equations (ODE). In this work, we consider settings where the underlying graph need not be a clique and is allowed to be suitably sparse. We show that if the minimum degree approaches infinity (however slowly) as the number of servers $N$ approaches infinity, and the ratio between the maximum degree and the minimum degree in each connected component approaches 1 uniformly, the occupancy process converges to the same system of ODE as the classical supermarket model. In particular, the asymptotic behavior of the occupancy process is insensitive to the precise network topology. We also study the case where the graph sequence is random, with the $N$-th graph given as an Erd\H{o}s-R\'enyi random graph on $N$ vertices with average degree $c(N)$. Annealed convergence of the occupancy process to the same deterministic limit is established under the condition $c(N)\to\infty$, and under a stronger condition $c(N)/\ln N\to\infty$, convergence (in probability) is shown for almost every realization of the random graph.

A classical result of Komlós, Sárközy, and Szemerédi states that every n‐vertex graph with minimum degree at least (1/2 + o(1))n contains every n‐vertex tree with maximum degree . Krivelevich, Kwan, and Sudakov proved that for every n‐vertex graph Gα with minimum degree at least αn for any fixed α > 0 and every n‐vertex tree T with bounded maximum degree, one can still find a copy of T in Gα with high probability after adding O(n) randomly chosen edges to Gα. We extend the latter results to trees with (essentially) unbounded maximum degree; for a given and α > 0, we determine up to a constant factor the number of random edges that we need to add to an arbitrary n‐vertex graph with minimum degree αn in order to guarantee with high probability a copy of any fixed n‐vertex tree with maximum degree at most Δ.

We say that a graph can be embedded into a graph if contains an isomorphic copy of as a subgraph. Guo and Volkmann conjectured that if is a connected graph with at least vertices and minimum degree at least , then any tree with vertices and maximum degree at most can be embedded into . In this paper, we give a result slightly stronger than this conjecture and obtain a sufficient and necessary condition that a tree with vertices and maximum degree at most can be embedded into a connected graph G with at least vertices and minimum degree at least . Our result implies that the conjecture of Guo and Volkmann is true with one exception. We also give an application to the Ramsey number of a tree versus a star.

As our main result, we prove that for every multigraph G = (V, E) which has no loops and is of order n, size m, and maximum degree $\Delta < {{{10}}^{-{{3}}}{{m}}\over \sqrt{{{8}}{{n}}}}$ there is a mapping ${{f}}:{{V}}\cup {{E}}\to \big\{{{1}},{{2}},\ldots,\big\lceil{{{m}}+{{2}}\over {{3}}}\big\rceil\big\}$ such that ${{f}}({{u}})+{{f}}({{uv}})+{{f}}({{v}})\not={{f}}({{u}}')+{{f}}({{u}}'{{v}}')+{{f}}({{v}}')$ for every ${{uv}},{{u}}'{{v}}'\in {{E}}$ with ${{uv}}\not={{u}}'{{v}}'$. Functions with this property were recently introduced and studied by Bača et al. and were called edge irregular total labelings. Our result confirms a recent conjecture of Ivančo and Jendrol′ about such labelings for dense graphs, for graphs where the maximum and minimum degree are not too different in terms of the order, and also for large graphs of bounded maximum degree. © 2008 Wiley Periodicals, Inc. J Graph Theory 57: 333–343, 2008

3-component domination numbers in graphs

For a graph , let be the number of spanning trees of with maximum degree at most . For , it is proved that every connected ‐vertex ‐regular graph with satisfies where approaches 1 extremely fast (e.g., ). The minimum degree requirement is essentially tight as for every there are connected ‐vertex ‐regular graphs with for which . Regularity may be relaxed, replacing with the geometric mean of the degree sequence and replacing with that also approaches 1, as long as the maximum degree is at most . The same holds with no restriction on the maximum degree as long as the minimum degree is at least .

In the vertex cover problem the input is a graph G and an integer k, and the goal is to decide whether there is a set of vertices S of size at most k such that every edge of G is incident on at least one vertex in S. We study the vertex cover problem on graphs with maximum degree 4 and minimum degree at least 2, parameterized by $$r = k-n/3$$ . We give an algorithm for this problem whose running time is $$O^*(1.6253^r)$$ . As a corollary, we obtain an $$O^*(1.2403^k)$$ -time algorithm for vertex cover on graphs with maximum degree 4.

We study the worst-case performance of the maximal matching heuristic applied to the Minimum Vertex Cover and Minimum Maximal Matching problems, through a careful analysis of tight examples. We show that the tight worst-case approximation ratio is asymptotic to \({\rm min}\, \{2, 1/(1-\sqrt{1-\epsilon})\}\) for graphs with an average degree at least εn and to min {2, 1/ε} for graphs with a minimum degree at least εn.KeywordsApproximation RatioMinimum DegreeVertex CoverMaximal MatchDense GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Given a simple graph $G$, denote by $\Delta(G)$, $\delta(G)$, and $\chi'(G)$ the maximum degree, the minimum degree, and the chromatic index of $G$, respectively. We say $G$ is $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ and $\chi'(H)\le \Delta(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $|E(G)|>\Delta(G) \lfloor |V(G)|/2 \rfloor$. Since a maximum matching in $G$ can have size at most $\lfloor |V(G)|/2 \rfloor$, it follows that $\chi'(G) = \Delta(G) +1$ if $G$ is overfull. Conversely, let $G$ be a $\Delta$-critical graph. The well known overfull conjecture of Chetwynd and Hilton asserts that $G$ is overfull provided $\Delta(G) > |V(G)|/3$. In this paper, we show that any $\Delta$-critical graph $G$ is overfull if $\Delta(G) - 7\delta(G)/4\ge (3|V(G)|-17)/4$.

This paper gives some bounds for the vertex degree function index Hf(G) in terms of the order and size of a graph G, where G is a simple, finite and connected graph with minimum degree δ and maximum degree ∆. Some families of graphs are also constructed to show that the bounds can be achieved.

This paper studies the relation between the connectivity and other parameters of a digraph (or graph), namely its order n, minimum degree δ, maximum degree Δ, diameter D, and a new parameter lpi;, 0 ≤ π ≤ δ − 2, related with the number of short paths (in the case of graphs l0 = ⌊(g − 1)/2⌋ where g stands for the girth). For instance, let G = (V,A) be a digraph on n vertices with maximum degree Δ and diameter D, so that n ≤ n(Δ, D) = 1 + Δ + Δ 2 + … + ΔD (Moore bound). As the main results it is shown that, if κ and λ denote respectively the connectivity and arc‐connectivity of G, equation image .Analogous results hold for graphs. © 1993 John Wiley & Sons, Inc.

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