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《Discrete Mathematics》2020,343(6):111712
The weak -coloring numbers of a graph were introduced by the first two authors as a generalization of the usual coloring number , and have since found interesting theoretical and algorithmic applications. This has motivated researchers to establish strong bounds on these parameters for various classes of graphs.Let denote the th power of . We show that, all integers and and graphs with satisfy ; for fixed tree width or fixed genus the ratio between this upper bound and worst case lower bounds is polynomial in . For the square of graphs , we also show that, if the maximum average degree , then . 相似文献
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《Applied and Computational Harmonic Analysis》2020,48(3):949-992
We propose a new method for performing multiscale analysis of functions defined on the vertices of a finite connected weighted graph. Our approach relies on a random spanning forest to downsample the set of vertices, and on approximate solutions of Markov intertwining relation to provide a subgraph structure and a filter bank leading to a wavelet basis of the set of functions. Our construction involves two parameters q and . The first one controls the mean number of kept vertices in the downsampling, while the second one is a tuning parameter between space localization and frequency localization. We provide an explicit reconstruction formula, bounds on the reconstruction operator norm and on the error in the intertwining relation, and a Jackson-like inequality. These bounds lead to recommend a way to choose the parameters q and . We illustrate the method by numerical experiments. 相似文献
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Two graphs G1 and G2 of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets do not intersect. Sauer and Spencer proved that if Δ (G1) Δ (G2) < 0.5n, then G1 and G2 pack.
In this note, we study an Ore-type analogue of the Sauer–Spencer Theorem. Let θ(G) = max{d(u) + d(v): uv∈E(G)}. We show that if θ(G1)Δ(G2) < n, then G1 and G2 pack. We also characterize the pairs (G1,G2) of n-vertex graphs satisfying θ(G1)Δ(G2) = n that do not pack.
This work was supported in part by NSF grant DMS-0400498. The work of the first author was also partly supported by grant
05-01-00816 of the Russian Foundation for Basic Research. 相似文献
109.
In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the ``rough' structure
of graphs excluding a fixed minor. This result was used to prove Wagner's Conjecture that finite graphs are well-quasi-ordered
under the graph minor relation. Recently, a number of beautiful results that use this structural result have appeared. Some
of these along with some other recent advances on graph minors are surveyed.
Research partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, Grant number
16740044, by Sumitomo Foundation, by C & C Foundation and by Inoue Research Award for Young Scientists
Supported in part by the Research Grant P1–0297 and by the CRC program
On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia 相似文献
110.
The slope-number of a graph G is the minimum number of distinct edge slopes in a straight-line drawing of G in the plane. We prove that for Δ5 and all large n, there is a Δ-regular n-vertex graph with slope-number at least . This is the best known lower bound on the slope-number of a graph with bounded degree. We prove upper and lower bounds on the slope-number of complete bipartite graphs. We prove a general upper bound on the slope-number of an arbitrary graph in terms of its bandwidth. It follows that the slope-number of interval graphs, cocomparability graphs, and AT-free graphs is at most a function of the maximum degree. We prove that graphs of bounded degree and bounded treewidth have slope-number at most . Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered. 相似文献