6‐Star‐Coloring of Subcubic Graphs

A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two adjacent vertices are assigned the same color) such that no path on four vertices is 2‐colored. The star chromatic number of G is the smallest integer k for which G admits a star coloring with k colors. In this paper, we prove that every subcubic graph is 6‐star‐colorable. Moreover, the upper bound 6 is best possible, based on the example constructed by Fertin, Raspaud, and Reed (J Graph Theory 47(3) (2004), 140–153).     

The Star Chromatic Numbers of Some Planar Graphs Derived from Wheels

The notion of the star chromatic number of a graph is a generalization of the chromatic number. In this paper, we calculate the star chromatic numbers of three infinite families of planar graphs. The first two families are derived from a 3-or 5-wheel by subdivisions, their star chromatic numbers being 2+2/(2n + 1), 2+3/(3n + 1), and 2+3(3n−1), respectively. The third family of planar graphs are derived from n odd wheels by Hajos construction with star chromatic numbers 3 + 1/n, which is a generalization of one result of Gao et al. Received September 21, 1998, Accepted April 9, 2001.     

Integral Domains Which Admit at Most Two Star Operations

In this article, we characterize domains which admit at most two star operations in the integrally closed and Noetherian cases. We also precisely count the number of star operations on an h-local Prüfer domain.     

A strong law of large numbers on the harmonic <Emphasis Type="Italic">p</Emphasis>-combination

In this paper, we establish a strong law of large numbers for the harmonic p-combinations of random star bodies. Starting from this theorem, we prove a strong law of large numbers in L _p space and provide the probabilistic version of dual Brunn-Minkowski inequality.     

Functorial Properties of Star Operations

We define a universal star operation to be an assignment *: A ? * _A of a star operation * _A on A to every integral domain A. Prime examples of universal star operations include the divisorial closure star operation v, the t-closure star operation t, and the star operation w = F _∞ of Hedstrom and Houston. For any universal star operation *, we say that an extension B ? A of integral domains is *-ideal class linked if there is a group homomorphism Cl_{* A} (A) → Cl_{* B} (B) of star class groups induced by the map I ? (IB)^{* _B} on the set of * _A -ideals I of A. We study several natural subclasses of the class of *-ideal class linked extensions.     

Note on list star edge-coloring of subcubic graphs

A star edge-coloring of a graph is a proper edge-coloring without bichromatic paths and cycles of length four. In this paper, we consider the list version of this coloring and prove that the list star chromatic index of every subcubic graph is at most 7, answering the question of Dvořák et al (J Graph Theory, 72 (2013), 313-326).     

Porosity and the L p -conjecture

Recently, Abtahi, Nasr-Isfahani, and Rejali [1] have shown that if G is a locally compact but not compact topological group and p > 2, then there are two functions, f and g, such that the convolution f*g{f\star g} is equal to ∞ on some set of positive measure. In the paper we show the nonexistence of f*g{f\star g} on a set of positive measure for (f, g) from a complement of a σ-porous set.     

On the sum relation of multiple Hurwitz zeta functions

In this paper we shall define the special-valued multiple Hurwitz zeta functions, namely the multiple t-values t(α) and define similarly the multiple star t-values as t^?(α). Then we consider the sum of all such multiple (star) t-values of fixed depth and weight with even argument and prove that such a sum can be evaluated when the evaluations of t({2m}ⁿ) and t*({2m}ⁿ) are clear. We give the evaluations of them in terms of the classical Euler numbers through their generating functions.     

Extremal cover times for random walks on trees

Let C_ν(T) denote the “cover time” of the tree T from the vertex v, that is, the expected number of steps before a random walk starting at v hits every vertex of T. Asymptotic lower bounds for C_ν(T) (for T a tree on n vertices) have been obtained recently by Kahn, Linial, Nisan and Saks, and by Devroye and Sbihi; here, we obtain the exact lower bound (approximately 2n In n) by showing that C_ν(T) is minimized when T is a star and v is one of its leaves. In addition, we show that the time to cover all vertices and then return to the starting point is minimized by a star (beginning at the center) and maximized by a path (beginning at one of the ends).     

Star coloring bipartite planar graphs

A star coloring of a graph is a proper vertex‐coloring such that no path on four vertices is 2‐colored. We prove that the vertices of every bipartite planar graph can be star colored from lists of size 14, and we give an example of a bipartite planar graph that requires at least eight colors to star color. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 1–10, 2009     

A new frequency‐uniform coercive boundary integral equation for acoustic scattering

A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L²(Γ) (where Γ is the surface of the scatterer) for all Lipschitz star‐shaped domains. Moreover, the coercivity is uniform in the wavenumber k = ω/c, where ω is the frequency and c is the speed of sound. The new boundary integral operator, which we call the “star‐combined” potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to implement as the standard one. Additionally, to the authors' knowledge, it is the only second‐kind integral operator for which convergence of the Galerkin method in L²(Γ) is proved without smoothness assumptions on Γ except that it is Lipschitz. The coercivity of the star‐combined operator implies frequency‐explicit error bounds for the Galerkin method for any approximation space. In particular, these error estimates apply to several hybrid asymptoticnumerical methods developed recently that provide robust approximations in the high‐frequency case. The proof of coercivity of the star‐combined operator critically relies on an identity first introduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains. © 2011 Wiley Periodicals, Inc.     

Graphs with Least Eigenvalue −2: The Star Complement Technique

Let G be a connected graph with least eigenvalue –2, of multiplicity k. A star complement for –2 in G is an induced subgraph H = G – X such that |X| = k and –2 is not an eigenvalue of H. In the case that G is a generalized line graph, a characterization of such subgraphs is used to decribe the eigenspace of –2. In some instances, G itself can be characterized by a star complement. If G is not a generalized line graph, G is an exceptional graph, and in this case it is shown how a star complement can be used to construct G without recourse to root systems.     

Proof of a conjecture of Alan Hartman

A tree T is said to be bad, if it is the vertex‐disjoint union of two stars plus an edge joining the center of the first star to an end‐vertex of the second star. A tree T is good, if it is not bad. In this article, we prove a conjecture of Alan Hartman that, for any spanning tree T of K_2m, where m ≥ 4, there exists a (2m − 1)‐edge‐coloring of K_2m such that all the edges of T receive distinct colors if and only if T is good. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 7–17, 1999     

On the Determination of Star and Convex Bodies by Section Functions

The i th section function of a star body in ⁿ gives the i -dimensional volumes of its sections by i -dimensional subspaces. It is shown that no star body is determined among all star bodies, up to reflection in the origin, by any of its i th section functions. Moreover, the set of star bodies that are determined among all star bodies, up to reflection in the origin, by their i th section functions for all i , is a nowhere dense set. The determination of convex bodies in this sense is also studied. The results complement and contrast with recent results on the determination of convex bodies by i th projection functions. The paper continues the development of the dual Brunn—Minkowski theory initiated by Lutwak. Received December 4, 1996, and in revised form April 14, 1997.     

Star coloring of sparse graphs

A proper coloring of the vertices of a graph is called a star coloring if the union of every two color classes induces a star forest. The star chromatic number χ_s(G) is the smallest number of colors required to obtain a star coloring of G. In this paper, we study the relationship between the star chromatic number χ_s(G) and the maximum average degree Mad(G) of a graph G. We prove that:

• 1. If G is a graph with , then χ_s(G)≤4.
• 2. If G is a graph with and girth at least 6, then χ_s(G)≤5.
• 3. If G is a graph with and girth at least 6, then χ_s(G)≤6.

These results are obtained by proving that such graphs admit a particular decomposition into a forest and some independent sets. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 201–219, 2009     

Simple transitive 2-representations of left cell 2-subcategories of projective functors for star algebras

In this article, we study simple transitive 2-representations of certain 2-subcategories of the 2-category of projective functors over a star algebra. We show that in the simplest case, which is associated to the Dynkin type A₂, simple transitive 2-representations are classified by cell 2-representations. In the general case we conjecture that there exist many simple transitive 2-representations which are not cell 2-representations and provide some evidence for our conjecture.     

Star discrepancy estimates for digital (t, m, 2)-nets and digital (t, 2) -sequences over Z2

Summary We prove upper bounds on the star discrepancy of digital (t, m, 2)-nets and (t, 2)-sequences over Z₂. The main tool is a decomposition lemma for digital (t, m, 2)-nets, which states that every digital (t, m, 2)-net is just the union of 2^tdigitally shifted digital (0, m - t, 2)-nets. Using this result we generalize upper bounds on the star discrepancy of digital (0, m, 2) -nets and (0, 2) -sequences.