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Register a new userMinimum degree conditions for containing an $r$-regular $r$-connected subgraph
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Added by
Max Hahn-Klimroth
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We study optimal minimum degree conditions when an $n$-vertex graph $G$ contains an $r$-regular $r$-connected subgraph. We prove for $r$ fixed and $n$ large the condition to be $delta(G) ge frac{n+r-2}{2}$ when $nr equiv 0 pmod 2$. This answers a question of M.~Kriesell.
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In this paper, we present a spectral sufficient condition for a graph to be Hamilton-connected in terms of signless Laplacian spectral radius with large minimum degree.
The weight of a subgraph $H$ in $G$ is the sum of the degrees in $G$ of vertices of $H$. The {em height} of a subgraph $H$ in $G$ is the maximum degree of vertices of $H$ in $G$. A star in a given graph is minor if its center has degree at most five in the given graph. Lebesgue (1940) gave an approximate description of minor $5$-stars in the class of normal plane maps with minimum degree five. In this paper, we give two descriptions of minor $5$-stars in plane graphs with minimum degree five. By these descriptions, we can extend several results and give some new results on the weight and height for some special plane graphs with minimum degree five.
An edge-coloured graph $G$ is called $properly$ $connected$ if every two vertices are connected by a proper path. The $proper$ $connection$ $number$ of a connected graph $G$, denoted by $pc(G)$, is the smallest number of colours that are needed in order to make $G$ properly connected. Susan A. van Aardt et al. gave a sufficient condition for the proper connection number to be at most $k$ in terms of the size of graphs. In this note, %optimizes the boundary of the number of edges %we study the $proper$ $connection$ $number$ is under the conditions of adding the minimum degree and optimizing the number of edges. our main result is the following, by adding a minimum degree condition: Let $G$ be a connected graph of order $n$, $kgeq3$. If $|E(G)|geq binom{n-m-(k+1-m)(delta+1)}{2} +(k+1-m)binom{delta+1}{2}+k+2$, then $pc(G)leq k$, where $m$ takes the value $t$ if $delta=1$ and $lfloor frac{k}{delta-1} rfloor$ if $deltageq2$. Furthermore, if $k=2$ and $delta=2$, %(i.e., $|E(G)|geq binom{n-5}{2} +7$) $pc(G)leq 2$, except $Gin {G_{1}, G_{n}}$ ($ngeq8$), where $G_{1}=K_{1}vee 3K_{2}$ and $G_{n}$ is obtained by taking a complete graph $K_{n-5}$ and $K_{1}vee (2K_{2}$) with an arbitrary vertex of $K_{n-5}$ and a vertex with $d(v)=4$ in $K_{1}vee (2K_{2}$) being joined. If $k=2$, $delta geq 3$, we conjecture $pc(G)leq 2$, where $m$ takes the value $1$ if $delta=3$ and $0$ if $deltageq4$ in the assumption.
Minimum $k$-Section denotes the NP-hard problem to partition the vertex set of a graph into $k$ sets of sizes as equal as possible while minimizing the cut width, which is the number of edges between these sets. When $k$ is an input parameter and $n$ denotes the number of vertices, it is NP-hard to approximate the width of a minimum $k$-section within a factor of $n^c$ for any $c<1$, even when restricted to trees with constant diameter. Here, we show that every tree $T$ allows a $k$-section of width at most $(k-1) (2 + 16n / diam(T) ) Delta(T)$. This implies a polynomial-time constant-factor approximation for the Minimum $k$-Section Problem when restricted to trees with linear diameter and constant maximum degree. Moreover, we extend our results from trees to arbitrary graphs with a given tree decomposition.
The minimum-weight $2$-edge-connected spanning subgraph (2-ECSS) problem is a natural generalization of the well-studied minimum-weight spanning tree (MST) problem, and it has received considerable attention in the area of network design. The latter problem asks for a minimum-weight subgraph with an edge connectivity of $1$ between each pair of vertices while the former strengthens this edge-connectivity requirement to $2$. Despite this resemblance, the 2-ECSS problem is considerably more complex than MST. While MST admits a linear-time centralized exact algorithm, 2-ECSS is NP-hard and the best known centralized approximation algorithm for it (that runs in polynomial time) gives a $2$-approximation. In this paper, we give a deterministic distributed algorithm with round complexity of $widetilde{O}(D+sqrt{n})$ that computes a $(5+epsilon)$-approximation of 2-ECSS, for any constant $epsilon>0$. Up to logarithmic factors, this complexity matches the $widetilde{Omega}(D+sqrt{n})$ lower bound that can be derived from Das Sarma et al. [STOC11], as shown by Censor-Hillel and Dory [OPODIS17]. Our result is the first distributed constant approximation for 2-ECSS in the nearly optimal time and it improves on a recent randomized algorithm of Dory [PODC18], which achieved an $O(log n)$-approximation in $widetilde{O}(D+sqrt{n})$ rounds. We also present an alternative algorithm for $O(log n)$-approximation, whose round complexity is linear in the low-congestion shortcut parameter of the network, following a framework introduced by Ghaffari and Haeupler [SODA16]. This algorithm has round complexity $widetilde{O}(D+sqrt{n})$ in worst-case networks but it provably runs much faster in many well-behaved graph families of interest. For instance, it runs in $widetilde{O}(D)$ time in planar networks and those with bounded genus, bounded path-width or bounded tree-width.
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