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1. CMB Online first

Vetrik, Tomas
 The metric dimension of circulant graphs A subset $W$ of the vertex set of a graph $G$ is called a resolving set of $G$ if for every pair of distinct vertices $u, v$ of $G$, there is $w \in W$ such that the~distance of $w$ and $u$ is different from the distance of $w$ and $v$. The~cardinality of a~smallest resolving set is called the metric dimension of $G$, denoted by $dim(G)$. The circulant graph $C_n (1, 2, \dots , t)$ consists of the vertices $v_0, v_1, \dots , v_{n-1}$ and the~edges $v_i v_{i+j}$, where $0 \le i \le n-1$, $1 \le j \le t$ $(2 \le t \le \lfloor \frac{n}{2} \rfloor)$, the indices are taken modulo $n$. Grigorious et al. [On the metric dimension of circulant and Harary graphs, Applied Mathematics and Computation 248 (2014), 47--54] proved that $dim(C_n (1,2, \dots , t)) \ge t+1$ for $t \lt \lfloor \frac{n}{2} \rfloor$, $n \ge 3$, and they presented a~conjecture saying that $dim(C_n (1,2, \dots , t)) = t+p-1$ for $n=2tk+t+p$, where $3 \le p \le t+1$. We disprove both statements. We show that if $t \ge 4$ is even, there exists an infinite set of values of $n$ such that $dim(C_n (1,2, \dots , t)) = t$. We also prove that $dim(C_n (1,2, \dots , t)) \le t + \frac{p}{2}$ for $n=2tk+t+p$, where $t$ and $p$ are even, $t \ge 4$, $2 \le p \le t$ and $k \ge 1$. Keywords:metric dimension, resolving set, circulant graph, Cayley graph, distanceCategories:05C35, 05C12

2. CMB Online first

Hashemi, Ebrahim; Amirjan, R.
 Zero-divisor graphs of Ore extensions over reversible rings Let $R$ be an associative ring with identity. First we prove some results about zero-divisor graphs of reversible rings. Then we study the zero-divisors of the skew power series ring $R[[x;\alpha]]$, whenever $R$ is reversible and $\alpha$-compatible. Moreover, we compare the diameter and girth of the zero-divisor graphs of $\Gamma(R)$, $\Gamma(R[x;\alpha,\delta])$ and $\Gamma(R[[x;\alpha]])$, when $R$ is reversible and $(\alpha,\delta)$-compatible. Keywords:zero-divisor graphs, reversible rings, McCoy rings, polynomial rings, power series ringsCategories:13B25, 05C12, 16S36

3. CMB Online first

Liu, Ye
 On chromatic functors and stable partitions of graphs The chromatic functor of a simple graph is a functorization of the chromatic polynomial. M. Yoshinaga showed that two finite graphs have isomorphic chromatic functors if and only if they have the same chromatic polynomial. The key ingredient in the proof is the use of stable partitions of graphs. The latter is shown to be closely related to chromatic functors. In this note, we further investigate some interesting properties of chromatic functors associated to simple graphs using stable partitions. Our first result is the determination of the group of natural automorphisms of the chromatic functor, which is in general a larger group than the automorphism group of the graph. The second result is that the composition of the chromatic functor associated to a finite graph restricted to the category $\mathrm{FI}$ of finite sets and injections with the free functor into the category of complex vector spaces yields a consistent sequence of representations of symmetric groups which is representation stable in the sense of Church-Farb. Keywords:chromatic functor, stable partition, representation stabilityCategories:05C15, 20C30

4. CMB Online first

 On the Diameter of Unitary Cayley Graphs of Rings The unitary Cayley graph of a ring $R$, denoted $\Gamma(R)$, is the simple graph defined on all elements of $R$, and where two vertices $x$ and $y$ are adjacent if and only if $x-y$ is a unit in $R$. The largest distance between all pairs of vertices of a graph $G$ is called the diameter of $G$, and is denoted by ${\rm diam}(G)$. It is proved that for each integer $n\geq1$, there exists a ring $R$ such that ${\rm diam}(\Gamma(R))=n$. We also show that ${\rm diam}(\Gamma(R))\in \{1,2,3,\infty\}$ for a ring $R$ with $R/J(R)$ self-injective and classify all those rings with ${\rm diam}(\Gamma(R))=1$, 2, 3 and $\infty$, respectively. Keywords:unitary Cayley graph, diameter, $k$-good, unit sum number, self-injective ringCategories:05C25, 16U60, 05C12

5. CMB Online first

Bouchemakh, Isma; Fatma, Kaci
 On the dual KÃ¶nig property of the order-interval hypergraph of two classes of N-free posets Let $P$ be a finite N-free poset. We consider the hypergraph $\mathcal{H}(P)$ whose vertices are the elements of $P$ and whose edges are the maximal intervals of $P$. We study the dual KÃ¶nig property of $\mathcal{H}(P)$ in two subclasses of N-free class. Keywords:poset, interval, N-free, hypergraph, KÃ¶nig property, dual KÃ¶nig propertyCategory:05C65

6. CMB Online first

Chen, Yichao; Yin, Xuluo
 The thickness of the Cartesian product of two graphs The thickness of a graph $G$ is the minimum number of planar subgraphs whose union is $G.$ A $t$-minimal graph is a graph of thickness $t$ which contains no proper subgraph of thickness $t.$ In this paper, upper and lower bounds are obtained for the thickness, $t(G\Box H)$, of the Cartesian product of two graphs $G$ and $H$, in terms of the thickness $t(G)$ and $t(H)$. Furthermore, the thickness of the Cartesian product of two planar graphs and of a $t$-minimal graph and a planar graph are determined. By using a new planar decomposition of the complete bipartite graph $K_{4k,4k},$ the thickness of the Cartesian product of two complete bipartite graphs $K_{n,n}$ and $K_{n,n}$ is also given, for $n\neq 4k+1$. Keywords:planar graph, thickness, Cartesian product, $t$-minimal graph, complete bipartite graphCategory:05C10

7. CMB Online first

 Some Results on the Annihilating-Ideal Graphs The annihilating-ideal graph of a commutative ring $R$, denoted by $\mathbb{AG}(R)$, is a graph whose vertex set consists of all non-zero annihilating ideals and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. Here, we show that if $R$ is a reduced ring and the independence number of $\mathbb{AG}(R)$ is finite, then the edge chromatic number of $\mathbb{AG}(R)$ equals its maximum degree and this number equals $2^{|{\rm Min}(R)|-1}-1$; also, it is proved that the independence number of $\mathbb{AG}(R)$ equals $2^{|{\rm Min}(R)|-1}$, where ${\rm Min}(R)$ denotes the set of minimal prime ideals of $R$. Then we give some criteria for a graph to be isomorphic with an annihilating-ideal graph of a ring. For example, it is shown that every bipartite annihilating-ideal graph is a complete bipartite graph with at most two horns. Among other results, it is shown that a finite graph $\mathbb{AG}(R)$ is not Eulerian, and it is Hamiltonian if and only if $R$ contains no Gorenstain ring as its direct summand. Keywords:annihilating-ideal graph, independence number, edge chromatic number, bipartite, cycleCategories:05C15, 05C69, 13E05, 13E10

8. CMB Online first

Akbari, Saieed; Miraftab, Babak; Nikandish, Reza
 Co-Maximal Graphs of Subgroups of Groups Let $H$ be a group. The co-maximal graph of subgroups of $H$, denoted by $\Gamma(H)$, is a graph whose vertices are non-trivial and proper subgroups of $H$ and two distinct vertices $L$ and $K$ are adjacent in $\Gamma(H)$ if and only if $H=LK$. In this paper, we study the connectivity, diameter, clique number and vertex chromatic number of $\Gamma(H)$. For instance, we show that if $\Gamma(H)$ has no isolated vertex, then $\Gamma(H)$ is connected with diameter at most $3$. Also, we characterize all finite groups whose co-maximal graphs are connected. Among other results, we show that if $H$ is a finitely generated solvable group and $\Gamma(H)$ is connected and moreover the degree of a maximal subgroup is finite, then $H$ is finite. Furthermore, we show that the degree of each vertex in the co-maximal graph of a general linear group over an algebraically closed field is zero or infinite. Keywords:co-maximal graphs of subgroups of groups, diameter, nilpotent group, solvable groupCategories:05C25, 05E15, 20D10, 20D15

9. CMB Online first

Dolžan, David
 The metric dimension of the total graph of a finite commutative ring We study the total graph of a finite commutative ring. We calculate its metric dimension in the case when the Jacobson radical of the ring is nontrivial and we examine the metric dimension of the total graph of a product of at most two fields, obtaining either exact values in some cases or bounds in other, depending on the number of elements in the respective fields. Keywords:total graph, finite ring, metric dimensionCategories:13M99, 05E40

10. CMB 2016 (vol 59 pp. 440)

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

11. CMB 2016 (vol 59 pp. 287)

Dukes, Peter; Lamken, E.R.; Ling, Alan C.H.
 An Existence Theory for Incomplete Designs An incomplete pairwise balanced design is equivalent to a pairwise balanced design with a distinguished block, viewed as a `hole'. If there are $v$ points, a hole of size $w$, and all (other) block sizes equal $k$, this is denoted IPBD$((v;w),k)$. In addition to congruence restrictions on $v$ and $w$, there is also a necessary inequality: $v \gt (k-1)w$. This article establishes two main existence results for IPBD$((v;w),k)$: one in which $w$ is fixed and $v$ is large, and the other in the case $v \gt (k-1+\epsilon) w$ when $w$ is large (depending on $\epsilon$). Several possible generalizations of the problem are also discussed. Keywords:block design, hypergraphCategory:05C70

12. CMB 2015 (vol 59 pp. 190)

Raeisi, Ghaffar; Zaghian, Ali
 Ramsey Number of Wheels Versus Cycles and Trees Let $G_1, G_2, \dots , G_t$ be arbitrary graphs. The Ramsey number $R(G_1, G_2, \dots, G_t)$ is the smallest positive integer $n$ such that if the edges of the complete graph $K_n$ are partitioned into $t$ disjoint color classes giving $t$ graphs $H_1,H_2,\dots,H_t$, then at least one $H_i$ has a subgraph isomorphic to $G_i$. In this paper, we provide the exact value of the $R(T_n,W_m)$ for odd $m$, $n\geq m-1$, where $T_n$ is either a caterpillar, a tree with diameter at most four or a tree with a vertex adjacent to at least $\lceil \frac{n}{2}\rceil-2$ leaves. Also, we determine $R(C_n,W_m)$ for even integers $n$ and $m$, $n\geq m+500$, which improves a result of Shi and confirms a conjecture of Surahmat et al. In addition, the multicolor Ramsey number of trees versus an odd wheel is discussed in this paper. Keywords:Ramsey number, wheel, tree, cycleCategories:05C15, 05C55, 05C65

13. CMB 2015 (vol 59 pp. 170)

Martínez-Pedroza, Eduardo
 A Note on Fine Graphs and Homological Isoperimetric Inequalities In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected $2$-complex $X$ with a linear homological isoperimetric inequality, a bound on the length of attaching maps of $2$-cells and finitely many $2$-cells adjacent to any edge must have a fine $1$-skeleton. We provide a positive answer to this question. We revisit a homological characterization of relative hyperbolicity, and show that a group $G$ is hyperbolic relative to a collection of subgroups $\mathcal P$ if and only if $G$ acts cocompactly with finite edge stabilizers on an connected $2$-dimensional cell complex with a linear homological isoperimetric inequality and $\mathcal P$ is a collection of representatives of conjugacy classes of vertex stabilizers. Keywords:isoperimetric functions, Dehn functions, hyperbolic groupsCategories:20F67, 05C10, 20J05, 57M60

14. CMB 2015 (vol 59 pp. 36)

Donovan, Diane M.; Griggs, Terry S.; McCourt, Thomas A.; Opršal, Jakub; Stanovský, David
 Distributive and Anti-distributive Mendelsohn Triple Systems We prove that the existence spectrum of Mendelsohn triple systems whose associated quasigroups satisfy distributivity corresponds to the Loeschian numbers, and provide some enumeration results. We do this by considering a description of the quasigroups in terms of commutative Moufang loops. In addition we provide constructions of Mendelsohn quasigroups that fail distributivity for as many combinations of elements as possible. These systems are analogues of Hall triple systems and anti-mitre Steiner triple systems respectively. Keywords:Mendelsohn triple system, quasigroup, distributive, Moufang loop, Loeschian numbersCategories:20N05, 05B07

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. CMB 2014 (vol 57 pp. 658)

Thang, Nguyen Tat
 Admissibility of Local Systems for some Classes of Line Arrangements Let $\mathcal{A}$ be a line arrangement in the complex projective plane $\mathbb{P}^2$ and let $M$ be its complement. A rank one local system $\mathcal{L}$ on $M$ is admissible if roughly speaking the cohomology groups $H^m(M,\mathcal{L})$ can be computed directly from the cohomology algebra $H^{*}(M,\mathbb{C})$. In this work, we give a sufficient condition for the admissibility of all rank one local systems on $M$. As a result, we obtain some properties of the characteristic variety $\mathcal{V}_1(M)$ and the Resonance variety $\mathcal{R}_1(M)$. Keywords:admissible local system, line arrangement, characteristic variety, multinet, resonance varietyCategories:14F99, 32S22, 52C35, 05A18, 05C40, 14H50

22. 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

23. CMB 2014 (vol 57 pp. 520)

Guo, Guangquan; Wang, Guoping
 Maximizing the Index of Trees with Given Domination Number The index of a graph $G$ is the maximum eigenvalue of its adjacency matrix $A(G)$. In this paper we characterize the extremal tree with given domination number that attains the maximum index. Keywords:trees, spectral radius, index, domination numberCategory:05C50

24. CMB 2013 (vol 57 pp. 72)

Grari, A.
 A Problem on Edge-magic Labelings of Cycles Kotzig and Rosa defined in 1970 the concept of edge-magic labelings as follows: let $G$ be a simple $(p,q)$-graph (that is, a graph of order $p$ and size $q$ without loops or multiple edges). A bijective function $f:V(G)\cup E(G)\rightarrow \{1,2,\ldots,p+q\}$ is an edge-magic labeling of $G$ if $f(u)+f(uv)+f(v)=k$, for all $uv\in E(G)$. A graph that admits an edge-magic labeling is called an edge-magic graph, and $k$ is called the magic sum of the labeling. An old conjecture of Godbold and Slater sets that all possible theoretical magic sums are attained for each cycle of order $n\ge 7$. Motivated by this conjecture, we prove that for all $n_0\in \mathbb{N}$, there exists $n\in \mathbb{N}$, such that the cycle $C_n$ admits at least $n_0$ edge-magic labelings with at least $n_0$ mutually distinct magic sums. We do this by providing a lower bound for the number of magic sums of the cycle $C_n$, depending on the sum of the exponents of the odd primes appearing in the prime factorization of $n$. Keywords:edge-magic, valence, $\otimes_h$-productCategory:05C78