Cops and Robber Game with a Fast Robber on Expander Graphs and Random Graphs

We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e., can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let ${c_{\infty}(G)}$ denote the number of cops needed to capture the robber in a graph G in this variant. We characterize graphs G with c _??(G) =? 1, and give an ${O( \mid V(G)\mid^2)}$ algorithm for their detection. We prove a lower bound for c _?? of expander graphs, and use it to prove three things. The first is that if ${np \geq 4.2 {\rm log}n}$ then the random graph ${G= \mathcal{G}(n, p)}$ asymptotically almost surely has ${\eta_{1}/p \leq \eta_{2}{\rm log}(np)/p}$ , for suitable positive constants ${\eta_{1}}$ and ${\eta_{2}}$ . The second is that a fixed-degree random regular graph G with n vertices asymptotically almost surely has ${c_{\infty}(G) = \Theta(n)}$ . The third is that if G is a Cartesian product of m paths, then ${n/4km^2 \leq c_{\infty}(G) \leq n/k}$ , where ${n = \mid V(G)\mid}$ and k is the number of vertices of the longest path.     

Graphs, partitions and classes of groups

If ${\mathfrak X}$ is a class of groups, Delizia et?al. (Bull Austral Math Soc 75:313–320, 2007) call a group G ${\mathfrak X}$ -transitive (or an ${\mathfrak XT}$ -group) if whenever ${\langle a,b\rangle}$ and ${\langle b,c\rangle}$ are in ${\mathfrak X} \langle a,c\rangle$ is also in ${\mathfrak X}$ ( ${a,b,c\in G}$ ). The structure of ${\mathfrak XT}$ -groups has been investigated for a number of classes of groups, by Delizia, Moravec and Nicotera and others. A graph can be associated with a group in many ways. Delizia, Moravec and Nicotera introduce a graph which is a generalisation of the commuting graph of a group, but do not make use of the graph. We will use the properties of the graph to investigate further classes of groups and to obtain more detailed structural information.     

The Majorization Theorem for Signless Laplacian Spectral Radii of Connected Graphs

Let ${\pi=(d_{1},d_{2},\ldots,d_{n})}$ and ${\pi'=(d'_{1},d'_{2},\ldots,d'_{n})}$ be two non-increasing degree sequences. We say ${\pi}$ is majorizated by ${\pi'}$ , denoted by ${\pi \vartriangleleft \pi'}$ , if and only if ${\pi\neq \pi'}$ , ${\sum_{i=1}^{n}d_{i}=\sum_{i=1}^{n}d'_{i}}$ , and ${\sum_{i=1}^{j}d_{i}\leq\sum_{i=1}^{j}d'_{i}}$ for all ${j=1,2,\ldots,n}$ . If there exists one connected graph G with ${\pi}$ as its degree sequence and ${c=(\sum_{i=1}^{n}d_{i})/2-n+1}$ , then G is called a c-cyclic graph and ${\pi}$ is called a c-cyclic degree sequence. Suppose ${\pi}$ is a non-increasing c-cyclic degree sequence and ${\pi'}$ is a non-increasing graphic degree sequence, if ${\pi \vartriangleleft \pi'}$ and there exists some t ${(2\leq t\leq n)}$ such that ${d'_{t}\geq c+1}$ and ${d_{i}=d'_{i}}$ for all ${t+1\leq i\leq n}$ , then the majorization ${\pi \vartriangleleft \pi'}$ is called a normal majorization. Let μ(G) be the signless Laplacian spectral radius, i.e., the largest eigenvalue of the signless Laplacian matrix of G. We use C _π to denote the class of connected graphs with degree sequence π. If ${G \in C_{\pi}}$ and ${\mu(G)\geq \mu(G')}$ for any other ${G'\in C_{\pi}}$ , then we say G has greatest signless Laplacian radius in C _π . In this paper, we prove that: Let π and π′ be two different non-increasing c-cyclic (c ≥ 0) degree sequences, G and G′ be the connected c-cyclic graphs with greatest signless Laplacian spectral radii in C _π and C _π', respectively. If ${\pi \vartriangleleft \pi'}$ and it is a normal majorization, then ${\mu(G) < \mu(G')}$ . This result extends the main result of Zhang (Discrete Math 308:3143–3150, 2008).     

Note on Group Distance Magic Graphs G[C 4]

A group distance magic labeling or a ${\mathcal{G}}$ -distance magic labeling of a graph G =  (V, E) with ${|V | = n}$ is a bijection f from V to an Abelian group ${\mathcal{G}}$ of order n such that the weight ${w(x) = \sum_{y\in N_G(x)}f(y)}$ of every vertex ${x \in V}$ is equal to the same element ${\mu \in \mathcal{G}}$ , called the magic constant. In this paper we will show that if G is a graph of order n =  2 ^p (2k + 1) for some natural numbers p, k such that ${\deg(v)\equiv c \mod {2^{p+1}}}$ for some constant c for any ${v \in V(G)}$ , then there exists a ${\mathcal{G}}$ -distance magic labeling for any Abelian group ${\mathcal{G}}$ of order 4n for the composition G[C ₄]. Moreover we prove that if ${\mathcal{G}}$ is an arbitrary Abelian group of order 4n such that ${\mathcal{G} \cong \mathbb{Z}_2 \times\mathbb{Z}_2 \times \mathcal{A}}$ for some Abelian group ${\mathcal{A}}$ of order n, then there exists a ${\mathcal{G}}$ -distance magic labeling for any graph G[C ₄], where G is a graph of order n and n is an arbitrary natural number.     

On Zero-Sum {\mathbb{Z}_k} -Magic Labelings of 3-Regular Graphs

Let G =  (V, E) be a finite loopless graph and let (A, +) be an abelian group with identity 0. Then an A-magic labeling of G is a function ${\phi}$ from E into A ? {0} such that for some ${a \in A, \sum_{e \in E(v)} \phi(e) = a}$ for every ${v \in V}$ , where E(v) is the set of edges incident to v. If ${\phi}$ exists such that a =  0, then G is zero-sum A-magic. Let zim(G) denote the subset of ${\mathbb{N}}$ (the positive integers) such that ${1 \in zim(G)}$ if and only if G is zero-sum ${\mathbb{Z}}$ -magic and ${k \geq 2 \in zim(G)}$ if and only if G is zero-sum ${\mathbb{Z}_k}$ -magic. We establish that if G is 3-regular, then ${zim(G) = \mathbb{N} - \{2\}}$ or ${\mathbb{N} - \{2,4\}.}$      

On the unique representation of very strong algebraic geometry codes

This paper addresses the question of retrieving the triple ${(\mathcal X,\mathcal P, E)}$ from the algebraic geometry code ${\mathcal C = \mathcal C_L(\mathcal X, \mathcal P, E)}$ , where ${\mathcal X}$ is an algebraic curve over the finite field ${\mathbb F_q, \,\mathcal P}$ is an n-tuple of ${\mathbb F_q}$ -rational points on ${\mathcal X}$ and E is a divisor on ${\mathcal X}$ . If ${\deg(E)\geq 2g+1}$ where g is the genus of ${\mathcal X}$ , then there is an embedding of ${\mathcal X}$ onto ${\mathcal Y}$ in the projective space of the linear series of the divisor E. Moreover, if ${\deg(E)\geq 2g+2}$ , then ${I(\mathcal Y)}$ , the vanishing ideal of ${\mathcal Y}$ , is generated by ${I_2(\mathcal Y)}$ , the homogeneous elements of degree two in ${I(\mathcal Y)}$ . If ${n >2 \deg(E)}$ , then ${I_2(\mathcal Y)=I_2(\mathcal Q)}$ , where ${\mathcal Q}$ is the image of ${\mathcal P}$ under the map from ${\mathcal X}$ to ${\mathcal Y}$ . These three results imply that, if ${2g+2\leq m < \frac{1}{2}n}$ , an AG representation ${(\mathcal Y, \mathcal Q, F)}$ of the code ${\mathcal C}$ can be obtained just using a generator matrix of ${\mathcal C}$ where ${\mathcal Y}$ is a normal curve in ${\mathbb{P}^{m-g}}$ which is the intersection of quadrics. This fact gives us some clues for breaking McEliece cryptosystem based on AG codes provided that we have an efficient procedure for computing and decoding the representation obtained.     

Toughness and Matching Extension in {\mathcal{P}_3}-Dominated Graphs

Let G be a connected graph. For ${x,y\in V(G)}$ with d(x, y) = 2, we define ${J(x,y)= \{u \in N(x)\cap N(y)\mid N[u] \subseteq N[x] \,{\cup}\,N[y] \}}$ and ${J'(x,y)= \{u \in N(x) \cap N(y)\,{\mid}\,{\rm if}\ v \in N(u){\setminus}(N[x] \,{\cup}\, N[y])\ {\rm then}\ N[x] \,{\cup}\, N[y]\,{\cup}\,N[u]{\setminus}\{x,y\}\subseteq N[v]\}}$ . A graph G is quasi-claw-free if ${J(x,y) \not= \emptyset}$ for each pair (x, y) of vertices at distance 2 in G. Broersma and Vumar (in Math Meth Oper Res. doi:10.1007/s00186-008-0260-7) introduced ${\mathcal{P}_{3}}$ -dominated graphs defined as ${J(x,y)\,{\cup}\, J'(x,y)\not= \emptyset}$ for each ${x,y \in V(G)}$ with d(x, y) = 2. This class properly contains that of quasi-claw-free graphs, and hence that of claw-free graphs. In this note, we prove that a 2-connected ${\mathcal{P}_3}$ -dominated graph is 1-tough, with two exceptions: K _2,3 and K _1,1,3, and prove that every even connected ${\mathcal{P}_3}$ -dominated graph ${G\ncong K_{1,3}}$ has a perfect matching. Moreover, we show that every even (2p + 1)-connected ${\mathcal{P}_3}$ -dominated graph is p-extendable. This result follows from a stronger result concerning factor-criticality of ${\mathcal{P}_3}$ -dominated graphs.     

On Suborbital Graphs for the Extended Modular Group {\hat{\Gamma}}

In this paper, we show that the extended modular group ${\hat{\Gamma}}$ acts on ${\hat{\mathbb{Q}}}$ transitively and imprimitively. Then the number of orbits of ${\hat{\Gamma} _{0}(N)}$ on ${\hat{\mathbb{Q}}}$ is calculated and compared with the number of orbits of ${\Gamma _{0}(N)}$ on ${\hat{\mathbb{Q}}}$ . Especially, we obtain the graphs ${\hat{G}_{u, N}}$ of ${\hat{\Gamma}_{0}(N)}$ on ${\hat{\mathbb{Q}}}$ , for each ${N\in\mathbb{N}}$ and each unit ${u \in U_{N} }$ , then we determine the suborbital graph ${\hat{F}_{u,N}}$ . We also give the edge conditions in ${\hat{G}_{u, N}}$ and the necessary and sufficient conditions for a circuit to be triangle in ${\hat{F}_{u, N}.}$      

Chromatic Gallai identities operating on Lovász number

If $G$ is a triangle-free graph, then two Gallai identities can be written as $\alpha (G)+\overline{\chi }(L(G))=|V(G)|=\alpha (L(G))+\overline{\chi }(G)$ , where $\alpha $ and $\overline{\chi }$ denote the stability number and the clique-partition number, and $L(G)$ is the line graph of  $G$ . We show that, surprisingly, both equalities can be preserved for any graph $G$ by deleting the edges of the line graph corresponding to simplicial pairs of adjacent arcs, according to any acyclic orientation of  $G$ . As a consequence, one obtains an operator $\Phi $ which associates to any graph parameter $\beta $ such that $\alpha (G) \le \beta (G) \le \overline{\chi }(G)$ for all graph $G$ , a graph parameter $\Phi _\beta $ such that $\alpha (G) \le \Phi _\beta (G) \le \overline{\chi }(G)$ for all graph $G$ . We prove that $\vartheta (G) \le \Phi _\vartheta (G)$ and that $\Phi _{\overline{\chi }_f}(G)\le \overline{\chi }_f(G)$ for all graph  $G$ , where $\vartheta $ is Lovász theta function and $\overline{\chi }_f$ is the fractional clique-partition number. Moreover, $\overline{\chi }_f(G) \le \Phi _\vartheta (G)$ for triangle-free $G$ . Comparing to the previous strengthenings $\Psi _\vartheta $ and $\vartheta ^{+ \triangle }$ of $\vartheta $ , numerical experiments show that $\Phi _\vartheta $ is a significant better lower bound for $\overline{\chi }$ than $\vartheta $ .     

Diagram Groupoids and von Neumann Algebras

The main purpose of this paper is to study certain algebraic structures induced by directed graphs. We have studied graph groupoids, which are algebraic structures induced by given graphs. By defining a certain groupoid-homomorphism ?? on the graph groupoid ${\mathbb{G}}$ of a given graph G, we define the diagram of G by the image ${\delta(\mathbb{G})}$ of ??, equipped with the inherited binary operation on ${\mathbb{G}}$ . We study the fundamental properties of the diagram ${\delta(\mathbb{G})}$ , and compare them with those of ${\mathbb{G}}$ . Similar to Cho (Acta Appl Math 95:95?C134, 2007), we construct the groupoid von Neumann algebra ${\mathcal{M}_{G}=vN(\delta(\mathbb{G}))}$ , generated by ${\delta(\mathbb{G})}$ , and consider the operator algebraic properties of ${\mathcal{M}_{G}}$ . In particular, we show ${\mathcal{M}_{G}}$ is *-isomorphic to a von Neumann algebra generated by a family of idempotent operators and nilpotent operators, under suitable representations.     

Atomic intersection of σ-fields and some of its consequences

Let ${(\Omega, \mathcal{F}, P)}$ be a probability space. For each ${\mathcal{G}\subset\mathcal{F}}$ , define ${\overline{\mathcal{G}}}$ as the σ-field generated by ${\mathcal{G}}$ and those sets ${F\in \mathcal{F}}$ satisfying ${P(F)\in\{0,1\}}$ . Conditions for P to be atomic on ${\cap_{i=1}^k\overline{\mathcal{A}_i}}$ , with ${\mathcal{A }_1,\ldots,\mathcal{A}_k\subset\mathcal{F}}$ sub-σ-fields, are given. Conditions for P to be 0-1-valued on ${\cap_{i=1}^k \overline{\mathcal{A}_i}}$ are given as well. These conditions are useful in various fields, including Gibbs sampling, iterated conditional expectations and the intersection property.     

The Linear t-Colorings of Sierpiński-Like Graphs

A proper t-coloring of a graph G is a mapping ${\varphi: V(G) \rightarrow [1, t]}$ such that ${\varphi(u) \neq \varphi(v)}$ if u and v are adjacent vertices, where t is a positive integer. The chromatic number of a graph G, denoted by ${\chi(G)}$ , is the minimum number of colors required in any proper coloring of G. A linear t-coloring of a graph is a proper t-coloring such that the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number of a graph G, denoted by ${lc(G)}$ , is the minimum t such that G has a linear t-coloring. In this paper, the linear t-colorings of Sierpiński-like graphs S(n, k), ${S^+(n, k)}$ and ${S^{++}(n, k)}$ are studied. It is obtained that ${lc(S(n, k))= \chi (S(n, k)) = k}$ for any positive integers n and k, ${lc(S^+(n, k)) = \chi(S^+(n, k)) = k}$ and ${lc(S^{++}(n, k)) = \chi(S^{++}(n, k)) = k}$ for any positive integers ${n \geq 2}$ and ${k \geq 3}$ . Furthermore, we have determined the number of paths and the length of each path in the subgraph induced by the union of any two color classes completely.     

Fractional f-Edge Cover Chromatic Index of Graphs

Let G be a graph, and let f be an integer function on V with ${1\leq f(v)\leq d(v)}$ to each vertex ${\upsilon \in V}$ . An f-edge cover coloring is a coloring of edges of E(G) such that each color appears at each vertex ${\upsilon \in V(G)}$ at least f(υ) times. The maximum number of colors needed to f-edge cover color G is called the f-edge cover chromatic index of G and denoted by ${\chi^{'}_{fc}(G)}$ . It is well known that any simple graph G has the f-edge cover chromatic index equal to δ _f (G) or δ _f (G) ? 1, where ${\delta_{f}(G)=\,min\{\lfloor \frac{d(v)}{f(v)} \rfloor: v\in V(G)\}}$ . The fractional f-edge cover chromatic index of a graph G, denoted by ${\chi^{'}_{fcf}(G)}$ , is the fractional f-matching number of the edge f-edge cover hypergraph ${\mathcal{H}}$ of G whose vertices are the edges of G and whose hyperedges are the f-edge covers of G. In this paper, we give an exact formula of ${\chi^{'}_{fcf}(G)}$ for any graph G, that is, ${\chi^{'}_{fcf}(G)=\,min \{\min\limits_{v\in V(G)}d_{f}(v), \lambda_{f}(G)\}}$ , where ${\lambda_{f}(G)=\min\limits_{S} \frac{|C[S]|}{\lceil (\sum\limits_{v\in S}{f(v)})/2 \rceil}}$ and the minimum is taken over all nonempty subsets S of V(G) and C[S] is the set of edges that have at least one end in S.     

Quasirandom permutations are characterized by 4-point densities

For permutations ${\pi}$ and ${\tau}$ of lengths ${|\pi|\le|\tau|}$ , let ${t(\pi,\tau)}$ be the probability that the restriction of ${\tau}$ to a random ${|\pi|}$ -point set is (order) isomorphic to ${\pi}$ . We show that every sequence ${\{\tau_j\}}$ of permutations such that ${|\tau_j|\to\infty}$ and ${t(\pi,\tau_j)\to 1/4!}$ for every 4-point permutation ${\pi}$ is quasirandom (that is, ${t(\pi,\tau_j)\to 1/|\pi|!}$ for every ${\pi}$ ). This answers a question posed by Graham.     

The Chvátal–Erdös Condition for Group Connectivity in Graphs

Let G be a graph and A an abelian group with the identity element 0 and ${|A| \geq 4}$ . Let D be an orientation of G. The boundary of a function ${f: E(G) \rightarrow A}$ is the function ${\partial f: V(G) \rightarrow A}$ given by ${\partial f(v) = \sum_{e \in E^+(v)}f(e) - \sum_{e \in E^-(v)}f(e)}$ , where ${v \in V(G), E^+(v)}$ is the set of edges with tail at v and ${E^-(v)}$ is the set of edges with head at v. A graph G is A-connected if for every b: V(G) → A with ${\sum_{v \in V(G)} b(v) = 0}$ , there is a function ${f: E(G) \mapsto A-\{0\}}$ such that ${\partial f = b}$ . A graph G is A-reduced to G′ if G′ can be obtained from G by contracting A-connected subgraphs until no such subgraph left. Denote by ${\kappa^{\prime}(G)}$ and α(G) the edge connectivity and the independent number of G, respectively. In this paper, we prove that for a 2-edge-connected simple graph G, if ${\kappa^{\prime}(G) \geq \alpha(G)-1}$ , then G is A-connected or G can be A-reduced to one of the five specified graphs or G is one of the 13 specified graphs.     

On additive decompositions of the set of primitive roots modulo p

It is conjectured that the set ${\mathcal {G}}$ of the primitive roots modulo p has no decomposition (modulo p) of the form ${\mathcal {G}= \mathcal {A} +\mathcal {B}}$ with ${|\mathcal {A}|\ge 2}$ , ${|\mathcal {B} |\ge 2}$ . This conjecture seems to be beyond reach but it is shown that if such a decomposition of ${\mathcal {G}}$ exists at all, then ${|\mathcal {A} |}$ , ${|\mathcal {B} |}$ must be around p ^1/2, and then this result is applied to show that ${\mathcal {G}}$ has no decomposition of the form ${\mathcal {G} =\mathcal {A} + \mathcal {B} + \mathcal {C}}$ with ${|\mathcal {A} |\ge 2}$ , ${|\mathcal {B} |\ge 2}$ , ${|\mathcal {C} |\ge 2}$ .     

Metrization of Weighted Graphs

We find a set of necessary and sufficient conditions under which the weight ${w: E \rightarrow \mathbb{R}^{+}}$ on the graph G = (V, E) can be extended to a pseudometric ${d : V \times V \rightarrow \mathbb{R}^{+}}$ . We describe the structure of graphs G for which the set ${\mathfrak{M}_{w}}$ of all such extensions contains a metric whenever w is strictly positive. Ordering ${\mathfrak{M}_{w}}$ by the pointwise order, we have found that the posets $({\mathfrak{M}_{w}, \leqslant)}$ contain the least elements ρ _0,w if and only if G is a complete k-partite graph with ${k \, \geqslant \, 2}$ . In this case the symmetric functions ${f : V \times V \rightarrow \mathbb{R}^{+}}$ , lying between ρ _0,w and the shortest-path pseudometric, belong to ${\mathfrak{M}_{w}}$ for every metrizable w if and only if the cardinality of all parts in the partition of V is at most two.     

Antibasis theorems for {\Pi^0_1} classes and the jump hierarchy

We prove two antibasis theorems for ${\Pi^0_1}$ classes. The first is a jump inversion theorem for ${\Pi^0_1}$ classes with respect to the global structure of the Turing degrees. For any ${P\subseteq 2^\omega}$ , define S(P), the degree spectrum of P, to be the set of all Turing degrees a such that there exists ${A \in P}$ of degree a. For any degree ${{\bf a \geq 0'}}$ , let ${\textrm{Jump}^{-1}({\bf a) = \{b : b' = a \}}}$ . We prove that, for any ${{\bf a \geq 0'}}$ and any ${\Pi^0_1}$ class P, if ${\textrm{Jump}^{-1} ({\bf a}) \subseteq S(P)}$ then P contains a member of every degree. For any degree ${{\bf a \geq 0'}}$ such that a is recursively enumerable (r.e.) in 0', let ${Jump_{\bf \leq 0'} ^{-1}({\bf a)=\{b : b \leq 0' \textrm{and} b' = a \}}}$ . The second theorem concerns the degrees below 0'. We prove that for any ${{\bf a\geq 0'}}$ which is recursively enumerable in 0' and any ${\Pi^0_1}$ class P, if ${\textrm{Jump}_{\bf \leq 0'} ^{-1}({\bf a)} \subseteq S(P)}$ then P contains a member of every degree.     

Circular Coloring of Planar Digraphs

Let D be a digraph. The circular chromatic number ${\chi_c(D)}$ and chromatic number ${\chi(D)}$ of D were proposed recently by Bokal et?al. Let ${\vec{\chi_c}(G)={\rm max}\{\chi_c(D)| D\, {\rm is\, an\, orientation\, of} G\}}$ . Let G be a planar graph and n?≥ 2. We prove that if the girth of G is at least ${\frac{10n-5}{3},}$ then ${\vec{\chi_c}(G)\leq \frac{n}{n-1}}$ . We also study the circular chromatic number of some special planar digraphs.