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1. CMB Online first

Dorfmeister, Josef F.; Inoguchi, Jun-ichi; Kobayashi, Shimpei
 On the Bernstein problem in the three-dimensional Heisenberg group In this note we present a simple alternative proof for the Bernstein problem in the three-dimensional Heisenberg group $\operatorname{Nil}_3$ by using the loop group technique. We clarify the geometric meaning of the two-parameter ambiguity of entire minimal graphs with prescribed Abresch-Rosenberg differential. Keywords:Bernstein problem, minimal graphs, Heisenberg group, loop groups, spinorsCategories:53A10, 53C42

2. CMB Online first

Gonçalves, Daniel; Li, Hui; Royer, Danilo
 Faithful representations of graph algebras via branching systems We continue to investigate branching systems of directed graphs and their connections with graph algebras. We give a sufficient condition under which the representation induced from a branching system of a directed graph is faithful and construct a large class of branching systems that satisfy this condition. We finish the paper by providing a proof of the converse of the Cuntz-Krieger uniqueness theorem for graph algebras by means of branching systems. Keywords:C*-algebra, graph algebra, Leavitt path algebra, branching system, representationCategories:46L05, 37A55

3. CMB Online first

Alfuraidan, Monther Rashed
 The Contraction Principle for Multivalued Mappings on a Modular Metric Space with a Graph We study the existence of fixed points for contraction multivalued mappings in modular metric spaces endowed with a graph. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. This paper can be seen as a generalization of Nadler's and Edelstein's fixed point theorems to modular metric spaces endowed with a graph. Keywords:fixed point theory, modular metric spaces, multivalued contraction mapping, connected digraph.Categories:47H09, 46B20, 47H10, 47E10

4. CMB 2015 (vol 58 pp. 610)

Rodger, C. A.; Whitt, Thomas Richard
 Path Decompositions of Kneser and Generalized Kneser Graphs Necessary and sufficient conditions are given for the existence of a graph decomposition of the Kneser Graph $KG_{n,2}$ and of the Generalized Kneser Graph $GKG_{n,3,1}$ into paths of length three. Keywords:Kneser graph, generalized Kneser graph, path decomposition, graph decompositionCategories:05C51, 05C70

5. CMB Online first

Zhang, Haihui
 A note on 3-choosability of planar graphs related to Montanssier's conjecture A graph $G=(V,E)$ is $L$-colorable if for a given list assignment $L=\{L(v):v\in V(G)\}$, there exists a proper coloring $c$ of $G$ such that $c(v)\in L(v)$ for all $v\in V$. If $G$ is $L$-colorable for every list assignment $L$ with $|L(v)|\geq k$ for all $v\in V$, then $G$ is said to be $k$-choosable. Montassier (Inform. Process. Lett. 99 (2006) 68-71) conjectured that every planar graph without cycles of length 4, 5, 6, is 3-choosable. In this paper, we prove that every planar graph without 5-, 6- and 10-cycles, and without two triangles at distance less than 3 is 3-choosable. Keywords:choosability, planar graph, cycleCategory:05C15

6. CMB 2015 (vol 58 pp. 320)

Llamas, Aurora; Martínez-Bernal, José
 Cover Product and Betti Polynomial of Graphs For disjoint graphs $G$ and $H$, with fixed vertex covers $C(G)$ and $C(H)$, their cover product is the graph $G \circledast H$ with vertex set $V(G)\cup V(H)$ and edge set $E(G)\cup E(H)\cup\{\{i,j\}:i\in C(G), j\in C(H)\}$. We describe the graded Betti numbers of $G\circledast H$ in terms of those of $G$ and $H$. As applications we obtain: (i) For any positive integer $k$ there exists a connected bipartite graph $G$ such that $\operatorname{reg} R/I(G)=\mu_S(G)+k$, where, $I(G)$ denotes the edge ideal of $G$, $\operatorname{reg} R/I(G)$ is the Castelnuovo--Mumford regularity of $R/I(G)$ and $\mu_S(G)$ is the induced or strong matching number of $G$; (ii) The graded Betti numbers of the complement of a tree only depends upon its number of vertices; (iii) The $h$-vector of $R/I(G\circledast H)$ is described in terms of the $h$-vectors of $R/I(G)$ and $R/I(H)$. Furthermore, in a different direction, we give a recursive formula for the graded Betti numbers of chordal bipartite graphs. Keywords:Castelnuovo--Mumford regularity, chordal bipartite graph, edge ideal, graded Betti number, induced matching number, monomial idealCategories:13D02, 05E45

7. CMB 2015 (vol 58 pp. 306)

Khoshkhah, Kaveh; Zaker, Manouchehr
 On the Largest Dynamic Monopolies of Graphs with a Given Average Threshold Let $G$ be a graph and $\tau$ be an assignment of nonnegative integer thresholds to the vertices of $G$. A subset of vertices, $D$ is said to be a $\tau$-dynamic monopoly, if $V(G)$ can be partitioned into subsets $D_0, D_1, \ldots, D_k$ such that $D_0=D$ and for any $i\in \{0, \ldots, k-1\}$, each vertex $v$ in $D_{i+1}$ has at least $\tau(v)$ neighbors in $D_0\cup \ldots \cup D_i$. Denote the size of smallest $\tau$-dynamic monopoly by $dyn_{\tau}(G)$ and the average of thresholds in $\tau$ by $\overline{\tau}$. We show that the values of $dyn_{\tau}(G)$ over all assignments $\tau$ with the same average threshold is a continuous set of integers. For any positive number $t$, denote the maximum $dyn_{\tau}(G)$ taken over all threshold assignments $\tau$ with $\overline{\tau}\leq t$, by $Ldyn_t(G)$. In fact, $Ldyn_t(G)$ shows the worst-case value of a dynamic monopoly when the average threshold is a given number $t$. We investigate under what conditions on $t$, there exists an upper bound for $Ldyn_{t}(G)$ of the form $c|G|$, where $c\lt 1$. Next, we show that $Ldyn_t(G)$ is coNP-hard for planar graphs but has polynomial-time solution for forests. Keywords:spread of influence in graphs, irreversible dynamic monopolies, target set selectionCategories:05C69, 05C85

8. CMB 2015 (vol 58 pp. 271)

 On Domination of Zero-divisor Graphs of Matrix Rings We study domination in zero-divisor graphs of matrix rings over a commutative ring with $1$. Keywords:vector space, linear transformation, zero-divisor graph, domination, local ringCategory:05C69

9. CMB 2015 (vol 58 pp. 317)

Lewkowicz, Marek Kazimierz
 Coloring Four-uniform Hypergraphs on Nine Vertices Every 4-uniform hypergraph on 9 vertices with at most 25 edges has property B. This gives the answer $m_9(4)=26$ to a question raised in 1968 by ErdÅs. Keywords:property B, coloring hypergraphsCategory:05C15

10. CMB 2014 (vol 58 pp. 105)

 On Graphs Associated with Character Degrees and Conjugacy Class Sizes of Direct Products of Finite Groups The prime vertex graph, $\Delta (X)$, and the common divisor graph, $\Gamma (X)$, are two graphs that have been defined on a set of positive integers $X$. Some properties of these graphs have been studied in the cases where either $X$ is the set of character degrees of a group or $X$ is the set of conjugacy class sizes of a group. In this paper, we gather some results on these graphs arising in the context of direct product of two groups. Keywords:prime vertex graph, common divisor graph, character degree, class sizes, graph operationCategories:20E45, 05C25, 05C76

11. CMB 2014 (vol 58 pp. 150)

Ostrovskii, Mikhail I.
 Connections Between Metric Characterizations of Superreflexivity and the Radon-NikodÃ½ Property for Dual Banach Spaces Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon-NikodÃ½m property (RNP) for dual spaces in terms of the infinite diamond. This paper is devoted to further study of relations between metric characterizations of superreflexivity and the RNP for dual spaces. The main result is that finite subsets of any set $M$ whose embeddability characterizes the RNP for dual spaces, characterize superreflexivity. It is also observed that the converse statement does not hold, and that $M=\ell_2$ is a counterexample. Keywords:Banach space, diamond graph, finite representability, metric characterization, Radon-NikodÃ½m property, superreflexivityCategories:46B85, 46B07, 46B22

12. CMB 2014 (vol 57 pp. 573)

Kiani, Sima; Maimani, Hamid Reza; Nikandish, Reza
 Some Results on the Domination Number of a Zero-divisor Graph In this paper, we investigate the domination, total domination and semi-total domination numbers of a zero-divisor graph of a commutative Noetherian ring. Also, some relations between the domination numbers of $\Gamma(R/I)$ and $\Gamma_I(R)$, and the domination numbers of $\Gamma(R)$ and $\Gamma(R[x,\alpha,\delta])$, where $R[x,\alpha,\delta]$ is the Ore extension of $R$, are studied. Keywords:zero-divisor graph, domination numberCategories:05C75, 13H10

13. CMB 2013 (vol 57 pp. 188)

 A Characterization of Bipartite Zero-divisor Graphs In this paper we obtain a characterization for all bipartite zero-divisor graphs of commutative rings $R$ with $1$, such that $R$ is finite or $|Nil(R)|\neq2$. Keywords:zero-divisor graph, bipartite graphCategories:13AXX, 05C25

14. CMB 2012 (vol 57 pp. 61)

Geschke, Stefan
 2-dimensional Convexity Numbers and $P_4$-free Graphs For $S\subseteq\mathbb R^n$ a set $C\subseteq S$ is an $m$-clique if the convex hull of no $m$-element subset of $C$ is contained in $S$. We show that there is essentially just one way to construct a closed set $S\subseteq\mathbb R^2$ without an uncountable $3$-clique that is not the union of countably many convex sets. In particular, all such sets have the same convexity number; that is, they require the same number of convex subsets to cover them. The main result follows from an analysis of the convex structure of closed sets in $\mathbb R^2$ without uncountable 3-cliques in terms of clopen, $P_4$-free graphs on Polish spaces. Keywords:convex cover, convexity number, continuous coloring, perfect graph, cographCategories:52A10, 03E17, 03E75

15. CMB 2011 (vol 56 pp. 317)

Dorais, François G.
 A Note on Conjectures of F. Galvin and R. Rado In 1968, Galvin conjectured that an uncountable poset $P$ is the union of countably many chains if and only if this is true for every subposet $Q \subseteq P$ with size $\aleph_1$. In 1981, Rado formulated a similar conjecture that an uncountable interval graph $G$ is countably chromatic if and only if this is true for every induced subgraph $H \subseteq G$ with size $\aleph_1$. TodorÄeviÄ has shown that Rado's Conjecture is consistent relative to the existence of a supercompact cardinal, while the consistency of Galvin's Conjecture remains open. In this paper, we survey and collect a variety of results related to these two conjectures. We also show that the extension of Rado's conjecture to the class of all chordal graphs is relatively consistent with the existence of a supercompact cardinal. Keywords:Galvin conjecture, Rado conjecture, perfect graph, comparability graph, chordal graph, clique-cover number, chromatic numberCategories:03E05, 03E35, 03E55

16. CMB 2011 (vol 56 pp. 265)

Chen, Yichao; Mansour, Toufik; Zou, Qian
 Embedding Distributions of Generalized Fan Graphs Total embedding distributions have been known for a few classes of graphs. Chen, Gross, and Rieper computed it for necklaces, close-end ladders and cobblestone paths. Kwak and Shim computed it for bouquets of circles and dipoles. In this paper, a splitting theorem is generalized and the embedding distributions of generalized fan graphs are obtained. Keywords:total embedding distribution, splitting theorem, generalized fan graphsCategory:05C10

17. CMB 2011 (vol 56 pp. 407)

 On Domination in Zero-Divisor Graphs We first determine the domination number for the zero-divisor graph of the product of two commutative rings with $1$. We then calculate the domination number for the zero-divisor graph of any commutative artinian ring. Finally, we extend some of the results to non-commutative rings in which an element is a left zero-divisor if and only if it is a right zero-divisor. Keywords:zero-divisor graph, dominationCategories:13AXX, 05C69

18. CMB 2009 (vol 53 pp. 378)

Zhou, Sizhong
 A New Sufficient Condition for a Graph To Be $(g,f,n)$-Critical Let $G$ be a graph of order $p$, let $a$, $b$, and $n$ be nonnegative integers with $1\leq a\lt b$, and let $g$ and $f$ be two integer-valued functions defined on $V(G)$ such that $a\leq g(x)\lt f(x)\leq b$ for all $x\in V(G)$. A $(g,f)$-factor of graph $G$ is a spanning subgraph $F$ of $G$ such that $g(x)\leq d_F(x)\leq f(x)$ for each $x\in V(F)$. Then a graph $G$ is called $(g,f,n)$-critical if after deleting any $n$ vertices of $G$ the remaining graph of $G$ has a $(g,f)$-factor. The binding number $\operatorname{bind}(G)$ of $G$ is the minimum value of ${|N_G(X)|}/{|X|}$ taken over all non-empty subsets $X$ of $V(G)$ such that $N_G(X)\neq V(G)$. In this paper, it is proved that $G$ is a $(g,f,n)$-critical graph if $\operatorname{bind}(G)\gt \frac{(a+b-1)(p-1)}{(a+1)p-(a+b)-bn+2} \quad\text{and}\quad p\geq \frac{(a+b-1)(a+b-2)}{a+1}+\frac{bn}{a}.$ Furthermore, it is shown that this result is best possible in some sense. Keywords:graph, $(g,f)$-factor, $(g,f,n)$-critical graph, binding numberCategory:05C70

19. CMB 2009 (vol 52 pp. 416)

Malik, Shabnam; Qureshi, Ahmad Mahmood; Zamfirescu, Tudor
 Hamiltonian Properties of Generalized Halin Graphs A Halin graph is a graph $H=T\cup C$, where $T$ is a tree with no vertex of degree two, and $C$ is a cycle connecting the end-vertices of $T$ in the cyclic order determined by a plane embedding of $T$. In this paper, we define classes of generalized Halin graphs, called $k$-Halin graphs, and investigate their Hamiltonian properties. Keywords:$k$-Halin graph, Hamiltonian, Hamiltonian connected, traceableCategories:05C45, 05C38

20. CMB 2009 (vol 52 pp. 257)

Ikeda, Toru
 Essential Surfaces in Graph Link Exteriors An irreducible graph manifold $M$ contains an essential torus if it is not a special Seifert manifold. Whether $M$ contains a closed essential surface of negative Euler characteristic or not depends on the difference of Seifert fibrations from the two sides of a torus system which splits $M$ into Seifert manifolds. However, it is not easy to characterize geometrically the class of irreducible graph manifolds which contain such surfaces. This article studies this problem in the case of graph link exteriors. Keywords:Graph link, Graph manifold, Seifert manifold, Essential surfaceCategory:57M25

21. CMB 2008 (vol 51 pp. 535)

Csorba, Péter
 On the Simple $\Z_2$-homotopy Types of Graph Complexes and Their Simple $\Z_2$-universality We prove that the neighborhood complex $\N(G)$, the box complex $\B(G)$, the homomorphism complex $\Hom(K_2,G)$and the Lov\'{a}sz complex $\L(G)$ have the same simple $\Z_2$-homotopy type in the sense of Whitehead. We show that these graph complexes are simple $\Z_2$-universal. Keywords:graph complexes, simple $\Z_2$-homotopy, universalityCategories:57Q10, 05C10, 55P10

22. CMB 2007 (vol 50 pp. 504)

Dukes, Peter; Ling, Alan C. H.
 Asymptotic Existence of Resolvable Graph Designs Let $v \ge k \ge 1$ and $\lam \ge 0$ be integers. A \emph{block design} $\BD(v,k,\lambda)$ is a collection $\cA$ of $k$-subsets of a $v$-set $X$ in which every unordered pair of elements from $X$ is contained in exactly $\lambda$ elements of $\cA$. More generally, for a fixed simple graph $G$, a \emph{graph design} $\GD(v,G,\lambda)$ is a collection $\cA$ of graphs isomorphic to $G$ with vertices in $X$ such that every unordered pair of elements from $X$ is an edge of exactly $\lambda$ elements of $\cA$. A famous result of Wilson says that for a fixed $G$ and $\lambda$, there exists a $\GD(v,G,\lambda)$ for all sufficiently large $v$ satisfying certain necessary conditions. A block (graph) design as above is \emph{resolvable} if $\cA$ can be partitioned into partitions of (graphs whose vertex sets partition) $X$. Lu has shown asymptotic existence in $v$ of resolvable $\BD(v,k,\lambda)$, yet for over twenty years the analogous problem for resolvable $\GD(v,G,\lambda)$ has remained open. In this paper, we settle asymptotic existence of resolvable graph designs. Keywords:graph decomposition, resolvable designsCategories:05B05, 05C70, 05B10

23. CMB 2007 (vol 50 pp. 460)

Spielberg, Jack
 Weak Semiprojectivity for Purely Infinite $C^*$-Algebras We prove that a separable, nuclear, purely infinite, simple $C^*$-algebra satisfying the universal coefficient theorem is weakly semiprojective if and only if its $K$-groups are direct sums of cyclic groups. Keywords:Kirchberg algebra, weak semiprojectivity, graph $C^*$-algebraCategories:46L05, 46L80, 22A22

24. CMB 2004 (vol 47 pp. 530)

Iranmanesh, A.; Khosravi, B.
 A Characterization of $PSU_{11}(q)$ Order components of a finite simple group were introduced in [4]. It was proved that some non-abelian simple groups are uniquely determined by their order components. As the main result of this paper, we show that groups $PSU_{11}(q)$ are also uniquely determined by their order components. As corollaries of this result, the validity of a conjecture of J. G. Thompson and a conjecture of W. Shi and J. Bi both on $PSU_{11}(q)$ are obtained. Keywords:Prime graph, order component, finite group,simple groupCategories:20D08, 20D05, 20D60

25. CMB 2003 (vol 46 pp. 122)

Moon, Myoungho
 On Certain Finitely Generated Subgroups of Groups Which Split Define a group $G$ to be in the class $\mathcal{S}$ if for any finitely generated subgroup $K$ of $G$ having the property that there is a positive integer $n$ such that $g^n \in K$ for all $g\in G$, $K$ has finite index in $G$. We show that a free product with amalgamation $A*_C B$ and an $\HNN$ group $A *_C$ belong to $\mathcal{S}$, if $C$ is in $\mathcal{S}$ and every subgroup of $C$ is finitely generated. Keywords:free product with amalgamation, $\HNN$ group, graph of groups, fundamental groupCategories:20E06, 20E08, 57M07
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