ISSN 1439-8516 CN 11-2039/O1
A star k-edge-coloring is a proper k-edge-coloring such that every connected bicolored subgraph is a path of length at most 3. The star chromatic index χ'st(G) of a graph G is the smallest integer k such that G has a star k-edge-coloring. The list star chromatic index ch'st(G) is defined analogously. The star edge coloring problem is known to be NP-complete, and it is even hard to obtain tight upper bound as it is unknown whether the star chromatic index for complete graph is linear or super linear. In this paper, we study, in contrast, the best linear upper bound for sparse graph classes. We show that for every ε > 0 there exists a constant c(ε) such that if mad(G) < 8/3-ε, then ch'st(G) ≤ 3Δ/2 +c(ε) and the coefficient 3/2 of Δ is the best possible. The proof applies a newly developed coloring extension method by assigning color sets with different sizes.
Given integer k and a k-graph F, let tk-1(n, F) be the minimum integer t such that every k-graph H on n vertices with codegree at least t contains an F -factor. For integers k ≥ 3 and 0 ≤ l ≤ k-1, let Yk,l be a k-graph with two edges that shares exactly l vertices. Han and Zhao (J. Combin. Theory Ser. A, (2015)) asked the following question:For all k ≥ 3, 0 ≤ l ≤ k-1 and sufficiently large n divisible by 2k -l, determine the exact value of tk-1(n, Yk,l). In this paper, we show that tk-1(n, Yk,l)=(n/2k -l) for k ≥ 3 and 1 ≤ l ≤ k-2, combining with two previously known results of Rödl, Ruciński and Szemerédi (J. Combin. Theory Ser. A, (2009)) and Gao, Han and Zhao (Combinatorics, Probability and Computing, (2019)), the question of Han and Zhao is solved completely.
We study the regularity of the solution of Dirichlet problem of Poisson equations over a bounded domain. A new sufficient condition, uniformly positive reach is introduced. Under the assumption that the closure of the underlying domain of interest has a uniformly positive reach, the H2 regularity of the solution of the Poisson equation is established. In particular, this includes all star-shaped domains whose closures are of positive reach, regardless if they are Lipschitz domains or non-Lipschitz domains. Application to the strong solution to the second order elliptic PDE in non-divergence form and the regularity of Helmholtz equations will be presented to demonstrate the usefulness of the new regularity condition.
The first and second Zagreb eccentricity indices of graph G are defined as:E1(G)=∑vi∈V(G) εG(vi)2, E2(G)=∑vivj∈E(G) εG(vi)εG(vj) where εG(vi) denotes the eccentricity of vertex vi in G. The eccentric complexity Cec(G) of G is the number of different eccentricities of vertices in G. In this paper we present some results on the comparison between (E1(G)/n) and (E2(G)/m) for any connected graphs G of order n with m edges, including general graphs and the graphs with given Cec. Moreover, a Nordhaus-Gaddum type result Cec(G) + Cec(G) is determined with extremal graphs at which the upper and lower bounds are attained respectively.
A class of second order non-autonomous Hamiltonian systems with asymptotically quadratic conditions is considered in this paper. Using Fountain Theorem, one multiplicity result of periodic solutions is obtained, which improves some previous results.
The purpose of this paper is to show that for one-sided symbolic systems, there exists an uncountable distributionally scrambled set contained in the set of proper positive upper Banach density recurrent points.
This paper is concerned with the travelling wavefronts of a nonlocal dispersal cooperation model with harvesting and state-dependent delay, which is assumed to be an increasing function of the population density with lower and upper bound. Especially, state-dependent delay is introduced into a nonlocal reaction-diffusion model. The conditions of Schauder's fixed point theorem are proved by constructing a reasonable set of functions Γ (see Section 2) and a pair of upper-lower solutions, so the existence of traveling wavefronts is established. The present study is continuation of a previous work that highlights the Laplacian diffusion.
In this article, we introduce a robust sparse test statistic which is based on the maximum type statistic. Both the limiting null distribution of the test statistic and the power of the test are analysed. It is shown that the test is particularly powerful against sparse alternatives. Numerical studies are carried out to examine the numerical performance of the test and to compare it with other tests available in the literature. The numerical results show that the test proposed significantly outperforms those tests in a range of settings, especially for sparse alternatives.
