We show that if $n=7k/i$ with $i \in \{1,2,3\}$ then the cop number of the generalised Petersen graph $GP(n,k)$ is $4$, with some small previously-known exceptions. It was previously proved by Ball et al. (2015) that the cop number of any generalised Petersen graph is at most $4$. The results in this paper explain all of the known generalised Petersen graphs that actually have cop number $4$ but were not previously explained by Morris et al. in a recent preprint, and places them in the context of infinite families. (More precisely, the preprint by Morris et al. explains all known generalised Petersen graphs with cop number $4$ and girth $8$, while this paper explains those that have girth $7$.)
A generalized Petersen graph $GP(n,k)$ is a regular cubic graph on $2n$ vertices (the parameter $k$ is used to define some of the edges). It was previously shown (Ball et al., 2015) that the cop number of $GP(n,k)$ is at most $4$, for all permissible values of $n$ and $k$. In this paper we prove that the cop number of "most" generalized Petersen graphs is exactly $4$. More precisely, we show that unless $n$ and $k$ fall into certain specified categories, then the cop number of $GP(n,k)$ is $4$. The graphs to which our result applies all have girth $8$. In fact, our argument is slightly more ge…
We offer a new, gradual approach to the largest girth problem for cubic graphs. It is easily observed that the largest possible girth of all $n$-vertex cubic graphs is attained by a $2$-connected graph $G=(V,E)$. By Petersen's graph theorem, $E$ is the disjoint union of a $2$-factor and a perfect matching $M$. We refer to the edges of $M$ as chords and classify the cycles in $G$ by their number of chords. We define $γ_k(n)$ to be the largest integer $g$ such that every cubic $n$-vertex graph with a given perfect matching $M$ has a cycle of length at most $g$ with at most $k$ chords. Here we de…
A hole is an induced cycle of length at least 4. Let $ł\ge 2$ be a positive integer, let ${\cal G}_l$ denote the family of graphs which have girth $2ł+1$ and have no holes of odd length at least $2ł+3$, and let $G\in {\cal G}_ł$. For a vertex $u\in V(G)$ and a nonempty set $S\subseteq V(G)$, let $d(u, S)=\min\{d(u, v):v\in S\}$, and let $L_i(S)=\{u\in V(G) \mbox{ and } d(u, S)=i\}$ for any integer $i\ge 0$. We show that if $G[S]$ is connected and $G[L_i(S)]$ is bipartite for each $i\in\{1, \ldots, \lfloor{ł\over 2}\rfloor\}$, then $G[L_i(S)]$ is bipartite for each $i>0$, and consequently $χ(G)…
The anti-Ramsey number $Ar(G,H)$ is the maximum number of colors in an edge-coloring of $G$ with no rainbow copy of $H$. In this paper, we determine the exact anti-Ramsey number in the generalized Petersen graph $P_{n,k}$ for cycles $C_d$, where $1\leq k\leq \lfloor \frac{n-1}{2} \rfloor$ and $5\le d \le 6$. We also give an algorithm to obtain the upper bound or lower bound of anti-Ramsey number.
For positive integers $s,t,u,v$, we define a bipartite graph $Γ_{\mathbb{R}}(X^s Y^t,X^u Y^v)$ where each partite set is a copy of $\mathbb{R}^3$, and a vertex $(a_1,a_2,a_3)$ in the first partite set is adjacent to a vertex $[x_1,x_2,x_3]$ in the second partite set if and only if \[ a_2 + x_2 = a_1^s x_1^t \quad \text{and} \quad a_3+x_3=a_1^ux_1^v. \] In this paper, we classify all such graphs according to girth.
One of Erdos's conjectures states that every triangle-free graph on $n$ vertices has an induced subgraph on $n/2$ vertices with at most $n^2/50$ edges. We report several partial results towards this conjecture. In particular, we establish the new bound $\frac{27}{1024}n^2$ on the number of edges in general case. We completely prove the conjecture for graphs of girth $\geq 5$, for graphs with independence number $\geq 2n/5$ and for strongly regular graphs. Each of these three classes includes both known (conjectured) extremal configurations, the 5-cycle and the Petersen graph.
Lower Bounds for the Cop Number When the Robber is Fast
我们考虑了一个变形的警察和强盗的游戏，其中强盗可以移动t边的时间，并表明，在这个变形中，d-正则图的周长大于2 t +2的警察数是Ω（d^t）。通过已知的笼阶的上界，这意味着当t>1时，连通n-顶点图的cop数可以大到Omega（n^{2/3}），当t>3时，可以大到Omega（n^{4/5}）。当1<t<7时，这改进了Frieze、Krivelevich和Loh（Variations on Cops and Robbers，J.图论，2011）的Omega（n^{（t-3）/（t-2）}）下界。我们还猜想了这个变体中的cop数的一般上界O（n^{t/t+1}），推广了Meyniel猜想.
Let $k,p,q$ be three positive integers. A graph $G$ with order $n$ is said to be $k$-placeable if there are $k$ edge disjoint copies of $G$ in the complete graph on $n$ vertices. A $(p,\,q)$-graph is a graph of order $p$ with $q$ edges. Packing results have proved useful in the study of the complexity of graph properties. Bollobás et al. investigated the $k$-placeable of $(n,\,n-2)$-graphs and $(n,\,n-1)$-graphs with $k=2$ and $k=3$. Motivated by their results, this paper characterizes $(n,\,n-1)$-graphs with girth at least $9$ which are $4$-placeable. We also consider the $k$-placeable of $(n…
For any field $\mathbb{F}$ and polynomials $f_{2},f_{3}\in\mathbb{F}[x,y]$, let $Γ_{\mathbb{F}}(f_{2},f_{3})$ denote the bipartite graph with vertex partition $P\cup L$, where $P$ and $L$ are two copies of $\mathbb{F}^{3}$, and $(p_{1},p_{2},p_{3})\in P$ is adjacent to $[l_{1},l_{2},l_{3}]\in L$ if and only if $p_{2}+l_{2}=f_{2}(p_{1},l_{1})$ and $p_{3}+l_{3}=f_{3}(p_{1},l_{1})$. The graph $Γ_{3}(\mathbb{F})=Γ_{\mathbb{F}}(xy,xy^{2})$ is known to be of girth eight. When $\mathbb{F}=\mathbb{F}_q$ is a finite field of odd size $q$ or $\mathbb{F}=\mathbb{F}_{\infty}$ is an algebraically closed fi…
We introduce the game of Surrounding Cops and Robbers on a graph, as a variant of the original game of Cops and Robbers. In contrast to the original game in which the cops win by occupying the same vertex as the robber, they now win by occupying each of the robber's neighbouring vertices. We denote by $σ(G)$ the {\em surrounding cop number} of $G$, namely the least number of cops required to surround a robber in the graph $G$. We present a number of results regarding this parameter, including general bounds as well as exact values for several classes of graphs. Particular classes of interest i…
A graph $G$ is $F$-saturated if it contains no copy of $F$ as a subgraph but the addition of any new edge to $G$ creates a copy of $F$. We prove that for $s \geq 3$ and $t \geq 2$, the minimum number of copies of $K_{1,t}$ in a $K_s$-saturated graph is $Θ( n^{t/2})$. More precise results are obtained when $t = 2$ where the problem is related to Moore graphs with diameter 2 and girth 5. We prove that for $s \geq 4$ and $t \geq 3$, the minimum number of copies of $K_{2,t}$ in an $n$-vertex $K_s$-saturated graph is at least $Ω( n^{t/5 + 8/5})$ and at most $O(n^{t/2 + 3/2})$. These results answer …
