The state-of-the-art modern pose-graph optimization (PGO) systems are vertex based. In this context the number of variables might be high, albeit cycles in graph (loop closures) is relatively low. For sparse problems particularly, cycle space has a significantly smaller dimension than vertices. By exploiting observation, paper we propose an alternative solution to PGO, that directly exploits sp...

Let $R$ be a ring (not necessarily commutative) with nonzero identity. We define $Gamma(R)$ to be the graph with vertex set $R$ in which two distinct vertices $x$ and $y$ are adjacent if and only if there exist unit elements $u,v$ of $R$ such that $x+uyv$ is a unit of $R$. In this paper, basic properties of $Gamma(R)$ are studied. We investigate connectivity and the girth of $Gamma(R)$, where $...

A cycle C in a graph G is extendible if there exists a cycle C 0 in G such that V (C) V (C 0) and jV (C 0)j = jV (C)j + 1. A graph G is cycle extendible if G contains at least one cycle and every nonhamiltonian cycle in G is extendible. A graph G is fully cycle extendible if G is cycle extendible and every vertex lies on a 3-cycle of G. G. Hendry asked if every hamiltonian chordal graph is full...

The Hamiltonian cycle problem in digraph is mapped into a matching cover bipartite graph. Based on this mapping, it is proved that determining existence a Hamiltonian cycle in graph is O(n). Abstract. Hamiltonian Cycle, Z-mapping graph, complexity, decision, matching covered, optimization Hamiltonian Cycle, Z-mapping graph, complexity, decision, matching covered, optimization

Let F, G and H be non-empty graphs. The notation F → (G,H) means that if any edge of F is colored by red or blue, then either the red subgraph of F con- tains a graph G or the blue subgraph of F contains a graph H. A graph F (without isolated vertices) is called a Ramsey (G,H)−minimal if F → (G,H) and for every e ∈ E(F), (F − e) 9 (G,H). The set of all Ramsey (G,H)−minimal graphs is denoted by ...

A Hamilton cycle is a cycle containing every vertex of a graph. A graph is called Hamiltonian if it contains a Hamilton cycle. The Hamilton cycle problem is to find the sufficient and necessary condition that a graph is Hamiltonian. In this paper, we give out some new kind of definitions of the subgraphs and determine the Hamiltoncity of edges according to the existence of the subgraphs in a gr...

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