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It is known that the cop number $c(G)$ of a connected graph $G$ can be bounded as a function of the genus of the graph $g(G)$. The best known bound, that $c(G) leq leftlfloor frac{3 g(G)}{2}rightrfloor + 3$, was given by Schr{o}der, who conjectured that in fact $c(G) leq g(G) + 3$. We give the first improvement to Schr{o}ders bound, showing that $c(G) leq frac{4g(G)}{3} + frac{10}{3}$.
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We show that the cop number of every generalized Petersen graph is at most 4. The strategy is to play a modified game of cops and robbers on an infinite cyclic covering space where the objective is to capture the robber or force the robber towards an end of the infinite graph. We prove that finite isometric subtrees are 1-guardable and apply this to determine the exact cop number of some families of generalized Petersen graphs. We also extend these ideas to prove that the cop number of any connected I-graph is at most 5.
An oriented graph $G^sigma$ is a digraph without loops or multiple arcs whose underlying graph is $G$. Let $Sleft(G^sigmaright)$ be the skew-adjacency matrix of $G^sigma$ and $alpha(G)$ be the independence number of $G$. The rank of $S(G^sigma)$ is called the skew-rank of $G^sigma$, denoted by $sr(G^sigma)$. Wong et al. [European J. Combin. 54 (2016) 76-86] studied the relationship between the skew-rank of an oriented graph and the rank of its underlying graph. In this paper, the correlation involving the skew-rank, the independence number, and some other parameters are considered. First we show that $sr(G^sigma)+2alpha(G)geqslant 2|V_G|-2d(G)$, where $|V_G|$ is the order of $G$ and $d(G)$ is the dimension of cycle space of $G$. We also obtain sharp lower bounds for $sr(G^sigma)+alpha(G),, sr(G^sigma)-alpha(G)$, $sr(G^sigma)/alpha(G)$ and characterize all corresponding extremal graphs.
We establish a lower bound for the energy of a complex unit gain graph in terms of the matching number of its underlying graph, and characterize all the complex unit gain graphs whose energy reaches this bound.
The $2$-cell embeddings of graphs on closed surfaces have been widely studied. It is well known that ($2$-cell) embedding a given graph $G$ on a closed orientable surface is equivalent to cyclically ordering the edges incident to each vertex of $G$. In this paper, we study the following problem: given a genus $g$ embedding $epsilon$ of the graph $G$ and a vertex of $G$, how many different ways of reembedding the vertex such that the resulting embedding $epsilon$ is of genus $g+Delta g$? We give formulas to compute this quantity and the local minimal genus achieved by reembedding. In the process we obtain miscellaneous results. In particular, if there exists a one-face embedding of $G$, then the probability of a random embedding of $G$ to be one-face is at least $prod_{ uin V(G)}frac{2}{deg( u)+2}$, where $deg( u)$ denotes the vertex degree of $ u$. Furthermore we obtain an easy-to-check necessary condition for a given embedding of $G$ to be an embedding of minimum genus.
In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph $G$ as a mapping $phicolon V(G)to mathbb{Z}$ such that for any two adjacent vertices $u$ and $v$ the colour $phi(u)$ is different from the colour $sigma(uv)phi(v)$, where is $sigma(uv)$ is the sign of the edge $uv$. The substantial part of Zaslavskys research concentrated on polynomial invariants related to signed graph colourings rather than on the behaviour of colourings of individual signed graphs. We continue the study of signed graph colourings by proposing the definition of a chromatic number for signed graphs which provides a natural extension of the chromatic number of an unsigned graph. We establish the basic properties of this invariant, provide bounds in terms of the chromatic number of the underlying unsigned graph, investigate the chromatic number of signed planar graphs, and prove an extension of the celebrated Brooks theorem to signed graphs.
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