|
Graphs with small total rainbow connection number
Yingbin MA, Lily CHEN, Hengzhe LI
Front. Math. China. 2017, 12 (4): 921-936.
https://doi.org/10.1007/s11464-017-0651-2
A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we investigate graphs with small total rainbow connection number. First, for a connected graph G, we prove that and Next, we investigate the total rainbow connection numbers of graphs G with diam and clique number . In this paper, we find Theorem 3 of [Discuss. Math. Graph Theory, 2011, 31(2): 313–320] is not completely correct, and we provide a complete result for this theorem.
References |
Related Articles |
Metrics
|
|
Neighbor sum distinguishing total chromatic number of K4-minor free graph
Hongjie SONG, Changqing XU
Front. Math. China. 2017, 12 (4): 937-947.
https://doi.org/10.1007/s11464-017-0649-9
A k-total coloring of a graph G is a mapping φ: V (G) ∪ E(G) →{1, 2, . . . , k} such that no two adjacent or incident elements in V (G) ∪ E(G) receive the same color. Let f(v) denote the sum of the color on the vertex v and the colors on all edges incident with v. We say that φ is a k-neighbor sum distinguishing total coloring of G if f(u) ≠ f(v) for each edge uv ∈ E(G). Denote the smallest value k in such a coloring of G. Pilśniak andWoźniak conjectured that for any simple graph with maximum degree Δ(G), . In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for K4-minor free graph G with Δ(G)≥5, if G contains no two adjacent Δ-vertices, otherwise, .
References |
Related Articles |
Metrics
|
|
Anisotropic weak Hardy spaces of Musielak-Orlicz type and their applications
Hui ZHANG, Chunyan QI, Baode LI
Front. Math. China. 2017, 12 (4): 993-1022.
https://doi.org/10.1007/s11464-016-0546-7
Anisotropy is a common attribute of the nature, which shows different characterizations in different directions of all or part of the physical or chemical properties of an object. The anisotropic property, in mathematics, can be expressed by a fairly general discrete group of dilations {Ak : k ∈ Z}, where A is a real n × n matrix with all its eigenvalues λ satisfy |λ|>1. The aim of this article is to study a general class of anisotropic function spaces, some properties and applications of these spaces. Let ϕ: Rn×[0,∞) →[0,∞) be an anisotropic p-growth function with p ∈ (0, 1]. The purpose of this article is to find an appropriate general space which includes weak Hardy space of Fefferman and Soria, weighted weak Hardy space of Quek and Yang, and anisotropic weak Hardy space of Ding and Lan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type and obtain its atomic characterization. As applications, we further obtain an interpolation theorem adapted to and the boundedness of the anisotropic Calderón-Zygmund operator from to . It is worth mentioning that the superposition principle adapted to the weak Musielak-Orlicz function space, which is an extension of a result of E. M. Stein, M. Taibleson and G. Weiss, plays an important role in the proofs of the atomic decomposition of and the interpolation theorem.
References |
Related Articles |
Metrics
|
15 articles
|