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Register a new userCharacterization of classes of graphs with large general position number
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Getting inspired by the famous no-three-in-line problem and by the general position subset selection problem from discrete geometry, the same is introduced into graph theory as follows. A set $S$ of vertices in a graph $G$ is a general position set if no element of $S$ lies on a geodesic between any two other elements of $S$. The cardinality of a largest general position set is the general position number ${rm gp}(G)$ of $G.$ In cite{ullas-2016} graphs $G$ of order $n$ with ${rm gp}(G)$ $in {2, n, n-1}$ were characterized. In this paper, we characterize the classes of all connected graphs of order $ngeq 4$ with the general position number $n-2.$
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A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number ${rm gp}(G)$ of $G$. It is proved that $Ssubseteq V(G)$ is in general position if and only if the components of $G[S]$ are complete subgraphs, the vertices of which form an in-transitive, distance-constant partition of $S$. If ${rm diam}(G) = 2$, then ${rm gp}(G)$ is the maximum of $omega(G)$ and the maximum order of an induced complete multipartite subgraph of the complement of $G$. As a consequence, ${rm gp}(G)$ of a cograph $G$ can be determined in polynomial time. If $G$ is bipartite, then ${rm gp}(G) leq alpha(G)$ with equality if ${rm diam}(G) in {2,3}$. A formula for the general position number of the complement of an arbitrary bipartite graph is deduced and simplified for the complements of trees, of grids, and of hypercubes.
We show that every n-vertex cubic graph with girth at least g have domination number at most 0.299871n+O(n/g)<3n/10+O(n/g).
Let Q(n,c) denote the minimum clique size an n-vertex graph can have if its chromatic number is c. Using Ramsey graphs we give an exact, albeit implicit, formula for the case c is at least (n+3)/2.
The Hadwiger number $h(G)$ is the order of the largest complete minor in $G$. Does sufficient Hadwiger number imply a minor with additional properties? In [2], Geelen et al showed $h(G)geq (1+o(1))ctsqrt{ln t}$ implies $G$ has a bipartite subgraph with Hadwiger number at least $t$, for some explicit $csim 1.276dotsc$. We improve this to $h(G) geq (1+o(1))tsqrt{log_2 t}$, and provide a construction showing this is tight. We also derive improved bounds for the topological minor variant of this problem.
Let $f$ be an optimal proper coloring of a graph $G$ and let $c$ be a coloring of the vertices of $G$ obtained by permuting the colors on vertices in the proper coloring $f$. The villainy of $c$, written $B(c)$, is the minimum number of vertices that must be recolored to obtain a proper coloring of $G$ with the additional condition that the number of times each color is used does not change. The villainy of $G$ is defined as $B(G)=max_{c}B(c)$, over all optimal proper colorings of $G$. In this paper, we characterize graphs $G$ with $B(G)=2$.
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