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.     

A witness version of the Cops and Robber game

The games considered are mixtures of Searching and Cops and Robber. The cops have partial information provided via witnesses who report “sightings” of the robber. The witnesses are able to provide information about the robber’s position but not the direction in which he is moving. The robber has perfect information. In the case when sightings occur at regular intervals, we present a recognition theorem for graphs on which a single cop suffices to guarantee a win. In a special case, this recognition theorem provides a characterization.     

A Note on the Game Chromatic Index of Graphs

In this note we prove that the game chromatic index χ _g ^′(G) of a graph G of arboricity k is at most Δ + 3k − 1. This improves a bound obtained by Cai and Zhu [J. Graph Theory 36 (2001), 144–155] for k-degenerate graphs. Tomasz Bartnicki: Research of the first author is supported by a PhD grant from Polish Ministry of Science and Higher Education N201 2128 33. Received: November 1, 2006. Final version received: December 22, 2007.     

The Game of Cops and Robbers on Directed Graphs with Forbidden Subgraphs

Acta Mathematicae Applicatae Sinica, English Series - The traditional game of cops and robbers is played on undirected graph. Recently, the same game played on directed graph is getting attention...     

Chasing a Fast Robber on Planar Graphs and Random Graphs

We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, that is, 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 denote the number of cops needed to capture the robber in a graph G in this variant, and let denote the treewidth of G. We show that if G is planar then , and there is a polynomial‐time constant‐factor approximation algorithm for computing . We also determine, up to constant factors, the value of of the Erd?s–Rényi random graph for all admissible values of p, and show that when the average degree is ω(1), is typically asymptotic to the domination number.     

Cops and Robbers on Graphs Based on Designs

We investigate the cop number of graphs based on combinatorial designs. Incidence graphs, point graphs, and block intersection graphs are studied, with an emphasis on finding families of graphs with large cop number. We generalize known results on Meyniel extremal families by supplying bounds on the incidence graphs of transversal designs, certain G‐designs, and BIBDs with Families of graphs with diameter 2, C₄‐free, and with unbounded chromatic number are described with the conjectured asymptotically maximum cop number.     

On the Edge Reconstruction of Graphs Embedded in Surfaces II

In this paper, we prove the following theorems. (i) Let G bea graph of minimum degree 5. If G is embeddable in a surface and satisfies (–5)|V(G)|+6()0, then G is edge reconstructible.(ii) Any graph of minimum degree 4 that triangulates a surfaceis edge reconstructible. (iii) Any graph which triangulatesa surface of characteristic 0 is edge reconstructible.     

Cops and Robbers on Planar‐Directed Graphs

Aigner and Fromme initiated the systematic study of the cop number of a graph by proving the elegant and sharp result that in every connected planar graph, three cops are sufficient to win a natural pursuit game against a single robber. This game, introduced by Nowakowski and Winkler, is commonly known as Cops and Robbers in the combinatorial literature. We extend this study to directed planar graphs, and establish separation from the undirected setting. We exhibit a geometric construction that shows that a sophisticated robber strategy can indefinitely evade three cops on a particular strongly connected planar‐directed graph.     

Enumeration of Non-Orientable 3-Manifolds Using Face-Pairing Graphs and Union-Find

Drawing together techniques from combinatorics and computer science, we improve the census algorithm for enumerating closed minimal P² 3-manifold triangulations. In particular, new constraints are proven for face-pairing graphs, and pruning techniques are improved using a modification of the union-find algorithm. Using these results we catalogue all 136 closed non-orientable P² 3-manifolds that can be formed from at most 10 tetrahedra.     

A Note on Cycle Lengths in Graphs

 We prove that for every c>0 there exists a constant K = K(c) such that every graph G with n vertices and minimum degree at least c n contains a cycle of length t for every even t in the interval [4,e c(G) − K] and every odd t in the interval [K,o c(G) − K], where e c(G) and o c(G) denote the length of the longest even cycle in G and the longest odd cycle in G respectively. We also give a rough estimate of the magnitude of K. Received: July 5, 2000 Final version received: April 17, 2002 2000 Mathematics Subject Classification. 05C38     

图的k部划分的一个注记

设k≥2是一个整数。本文证明了任意有m条边的图都存在一个顶点的划分V_1,V_2…,V_k,使得e(V_1,V_2…,V_k)≥k-1/k m+k-1/2k((2m+1/4)~1/2-1/2)-(k-2)~2/8k,且max{e(V_i):1≤i≤k}≤m/k~2+(k-1)/2k~2((2m+1/4)~1/2-1/2+3/8-7k-4/8k~2.我们的结果改进了[Fan G.,Hou J.,Zeng Q.,A bound for judicious k-partitions of graphs,Discrete Appl.Math.,2014,179:86—99]的主要结论.     

Gauss-Bonnet Formula, Finiteness Condition, and Characterizations of Graphs Embedded in Surfaces

Let G be an infinite graph embedded in a closed 2-manifold, such that each open face of the embedding is homeomorphic to an open disk and is bounded by finite number of edges. For each vertex x of G, define the combinatorial curvature

A Note on Uniquely lntersectable Graphs

The following conjecture of Alter and Wang is proven. Consider the intersection graph G_n,m,n?2m, determined by the family of all m-element subsets of an n-element set. Then any realization of G_n,m as an intersection graph by a family of sets satisfies |∪_iA_i|?n; and if |∪_iA_i|=n, then F must be the family of all m-element subsets of ∪_iA_i.     

A Note on Almost Regular Graphs

It can easily be seen that a conjecture of Runge does not hold for a class of graphs whose members will be called “almost regular”. This conjecture is replaced by a weaker one, and a classification of almost regular graphs is given.     

A Note on Semi-coloring of Graphs

Semi-coloring is a new type of edge coloring of graphs. In this note, we show that every graph has a semi-coloring. This answers a problem, posed by Daniely and Linial, in affirmative. It implies that every r-regular graph has at least ${\lceil\frac{r}{2}\rceil}$ different {K ₂, C _i | i ≥ 3}-factors.     

单圈图的零度的一个注记

The number of zero eigenvalues in the spectrum of the graph G is called its nullity and is denoted by η(G).In this paper,we determine the all extremal unicyclic graphs achieving the fifth upper bound n-6 and the sixth upperbound n-7.