Bounds on the length of a game of Cops and Robbers

In the game of Cops and Robbers, a team of cops attempts to capture a robber on a graph $G$. All players occupy vertices of $G$. The game operates in rounds; in each round the cops move to neighboring vertices, after which the robber does the same. The minimum number of cops needed to guarantee capture of a robber on $G$ is the cop number of $G$, denoted $c\left(G\right)$, and the minimum number of rounds needed for them to do so is the capture time. It has long been known that the capture time of an $n$-vertex graph with cop number $k$ is $O\left(n^{k+1}\right)$. More recently, Bonato et al. (2009) and Gaven?iak (2010) showed that for $k=1$, this upper bound is not asymptotically tight: for graphs with cop number 1, the cop can always win within $n?4$ rounds. In this paper, we show that the upper bound is tight when $k\ge 2$: for fixed $k\ge 2$, we construct arbitrarily large graphs $G$ having capture time at least ${\left(\frac{\left|V\left(G\right)\right|}{40k^4}\right)}^{k+1}$.In the process of proving our main result, we establish results that may be of independent interest. In particular, we show that the problem of deciding whether $k$ cops can capture a robber on a directed graph is polynomial-time equivalent to deciding whether $k$ cops can capture a robber on an undirected graph. As a corollary of this fact, we obtain a relatively short proof of a major conjecture of Goldstein and Reingold (1995), which was recently proved through other means (Kinnersley, 2015). We also show that $n$-vertex strongly-connected directed graphs with cop number 1 can have capture time $\Omega \left(n^2\right)$, thereby showing that the result of Bonato et al. (2009) does not extend to the directed setting.     

A note on degree sum conditions for 2-factors with a prescribed number of cycles in bipartite graphs

Let $G$ be a balanced bipartite graph of order $2n\ge 4$, and let ${\sigma }_{1,1}\left(G\right)$ be the minimum degree sum of two non-adjacent vertices in different partite sets of $G$. In 1963, Moon and Moser proved that if ${\sigma }_{1,1}\left(G\right)\ge n+1$, then $G$ is hamiltonian. In this note, we show that if $k$ is a positive integer, then the Moon–Moser condition also implies the existence of a 2-factor with exactly $k$ cycles for sufficiently large graphs. In order to prove this, we also give a ${\sigma }_{1,1}$ condition for the existence of $k$ vertex-disjoint alternating cycles with respect to a chosen perfect matching in $G$.     

The NLC-width and clique-width for powers of graphs of bounded tree-width

The $k$-power graph of a graph $G$ is a graph with the same vertex set as $G$, in that two vertices are adjacent if and only if, there is a path between them in $G$ of length at most $k$. A $k$-tree-power graph is the $k$-power graph of a tree, a $k$-leaf-power graph is the subgraph of some $k$-tree-power graph induced by the leaves of the tree.We show that (1) every $k$-tree-power graph has NLC-width at most $k+2$ and clique-width at most $k+2+max\{?\frac{k}{2}??1,0\}$, (2) every $k$-leaf-power graph has NLC-width at most $k$ and clique-width at most $k+max\{?\frac{k}{2}??2,0\}$, and (3) every $k$-power graph of a graph of tree-width $l$ has NLC-width at most ${(k+1)}^{l+1}?1$, and clique-width at most $2?{(k+1)}^{l+1}?2$.     

More odd graph theory from another point of view

The Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of $\left[n\right]=\{1,2,\dots ,n\}$ and an edge between any two vertices that are disjoint. If $n=2k+1$, then $K(n,k)$ is called an odd graph. Let $n>4$ and $1<k<\frac{n}{2}$. In the present paper, we show that if the Kneser graph $K(n,k)$ is of even order where $n$ is an odd integer or both of the integers $n,k$ are even, then $K(n,k)$ is a vertex-transitive non Cayley graph. Although, these are special cases of Godsil [7], unlike his proof that uses some very deep group-theoretical facts, ours uses no heavy group-theoretic facts. We obtain our results by using some rather elementary facts of number theory and group theory. We show that ‘almost all’ odd graphs are of even order, and consequently are vertex-transitive non Cayley graphs. Finally, we show that if $k>4$ is an even integer such that $k$ is not of the form $k=2^t$ for some $t>2$, then the line graph of the odd graph $O_{k+1}$ is a vertex-transitive non Cayley graph.     

Sparse graphs of girth at least five are packable

A graph is packable if it is a subgraph of its complement. The following statement was conjectured by Faudree, Rousseau, Schelp and Schuster in 1981: every non-star graph $G$ with girth at least $5$ is packable.The conjecture was proved by Faudree et al. with the additional condition that $G$ has at most $\frac{6}{5}n?2$ edges. In this paper, for each integer $k\ge 3$, we prove that every non-star graph with girth at least $5$ and at most $\frac{2k?1}{k}n?{\alpha }_k\left(n\right)$ edges is packable, where ${\alpha }_k\left(n\right)$ is $o\left(n\right)$ for every $k$. This implies that the conjecture is true for sufficiently large planar graphs.     

On 3-stable number conditions in n-connected claw-free graphs

For a subgraph $X$ of $G$, let ${\alpha }_G^3\left(X\right)$ be the maximum number of vertices of $X$ that are pairwise distance at least three in $G$. In this paper, we prove three theorems. Let $n$ be a positive integer, and let $H$ be a subgraph of an $n$-connected claw-free graph $G$. We prove that if $n\ge 2$, then either $H$ can be covered by a cycle in $G$, or there exists a cycle $C$ in $G$ such that ${\alpha }_G^3(H?V\left(C\right))\le {\alpha }_G^3\left(H\right)?n$. This result generalizes the result of Broersma and Lu that $G$ has a cycle covering all the vertices of $H$ if ${\alpha }_G^3\left(H\right)\le n$. We also prove that if $n\ge 1$, then either $H$ can be covered by a path in $G$, or there exists a path $P$ in $G$ such that ${\alpha }_G^3(H?V\left(P\right))\le {\alpha }_G^3\left(H\right)?n?1$. By using the second result, we prove the third result. For a tree $T$, a vertex of $T$ with degree one is called a leaf of $T$. For an integer $k\ge 2$, a tree which has at most $k$ leaves is called a $k$-ended tree. We prove that if ${\alpha }_G^3\left(H\right)\le n+k?1$, then $G$ has a $k$-ended tree covering all the vertices of $H$. This result gives a positive answer to the conjecture proposed by Kano et al. (2012).     

Stability in the Erdős–Gallai Theorem on cycles and paths,II

The Erd?s–Gallai Theorem states that for $k\ge 3$, any $n$-vertex graph with no cycle of length at least $k$ has at most $\frac{1}{2}(k?1)(n?1)$ edges. A stronger version of the Erd?s–Gallai Theorem was given by Kopylov: If $G$ is a 2-connected $n$-vertex graph with no cycle of length at least $k$, then $e\left(G\right)\le max\{h(n,k,2),h(n,k,?\frac{k?1}{2}?)\}$, where $h(n,k,a)?\left(\genfrac{}{}{0ex}{}{k?a}{2}\right)+a(n?k+a)$. Furthermore, Kopylov presented the two possible extremal graphs, one with $h(n,k,2)$ edges and one with $h(n,k,?\frac{k?1}{2}?)$ edges.In this paper, we complete a stability theorem which strengthens Kopylov’s result. In particular, we show that for $k\ge 3$ odd and all $n\ge k$, every $n$-vertex 2-connected graph $G$ with no cycle of length at least $k$ is a subgraph of one of the two extremal graphs or $e\left(G\right)\le max\{h(n,k,3),h(n,k,\frac{k?3}{2})\}$. The upper bound for $e\left(G\right)$ here is tight.     

List star chromatic index of sparse graphs

A star edge-coloring of a graph $G$ is a proper edge coloring such that every 2-colored connected subgraph of $G$ is a path of length at most 3. For a graph $G$, let the list star chromatic index of $G$, $c{h}_s^{\prime }\left(G\right)$, be the minimum $k$ such that for any $k$-uniform list assignment $L$ for the set of edges, $G$ has a star edge-coloring from $L$. Dvo?ák et al. (2013) asked whether the list star chromatic index of every subcubic graph is at most 7. In Kerdjoudj et al. (2017) we proved that it is at most 8. In this paper we consider graphs with any maximum degree, we proved that if the maximum average degree of a graph $G$ is less than $\frac{14}{5}$ (resp. 3), then $c{h}_s^{\prime }\left(G\right)\le 2\Delta \left(G\right)+2$ (resp. $c{h}_s^{\prime }\left(G\right)\le 2\Delta \left(G\right)+3$).     

Sums of powers of Catalan triangle numbers

In this paper, we consider combinatorial numbers ${\left(C_{m,k}\right)}_{m\ge 1,k\ge 0}$, mentioned as Catalan triangle numbers where $C_{m,k}?\left(\genfrac{}{}{0ex}{}{m?1}{k}\right)?\left(\genfrac{}{}{0ex}{}{m?1}{k?1}\right)$. These numbers unify the entries of the Catalan triangles $B_{n,k}$ and $A_{n,k}$ for appropriate values of parameters $m$ and $k$, i.e., $B_{n,k}=C_{2n,n?k}$ and $A_{n,k}=C_{2n+1,n+1?k}$. In fact, these numbers are suitable rearrangements of the known ballot numbers and some of these numbers are the well-known Catalan numbers $C_n$ that is $C_{2n,n?1}=C_{2n+1,n}=C_n$.We present identities for sums (and alternating sums) of $C_{m,k}$, squares and cubes of $C_{m,k}$ and, consequently, for $B_{n,k}$ and $A_{n,k}$. In particular, one of these identities solves an open problem posed in Gutiérrez et al. (2008). We also give some identities between ${\left(C_{m,k}\right)}_{m\ge 1,k\ge 0}$ and harmonic numbers ${\left(H_n\right)}_{n\ge 1}$. Finally, in the last section, new open problems and identities involving ${\left(C_n\right)}_{n\ge 0}$ are conjectured.     

A new approach to the chromatic number of the square of Kneser graph K(2k+1,k)

The vertices of Kneser graph $K(n,k)$ are the subsets of $\{1,2,\dots ,n\}$ of cardinality $k$, two vertices are adjacent if and only if they are disjoint. The square $G^2$ of a graph $G$ is defined on the vertex set of $G$ with two vertices adjacent if their distance in $G$ is at most 2. Z. Füredi, in 2002, proposed the problem of determining the chromatic number of the square of the Kneser graph. The first non-trivial problem arises when $n=2k+1$. It is believed that $\chi \left(K^2(2k+1,k)\right)=2k+c$ where $c$ is a constant, and yet the problem remains open. The best known upper bounds are by Kim and Park: $8k∕3+20∕3$ for 1$k\ge 3$ (Kim and Park, 2014) and $32k∕15+32$ for $k\ge 7$ (Kim and Park, 2016). In this paper, we develop a new approach to this coloring problem by employing graph homomorphisms, cartesian products of graphs, and linear congruences integrated with combinatorial arguments. These lead to $\chi \left(K^2(2k+1,k)\right)\le 5k∕2+c$, where $c$ is a constant in $\{5∕2,9∕2,5,6\}$, depending on $k\ge 2$.