Conceptual configuration design of line-foldable deployable space truss structures employing graph theory

From the perspective of the truss as a whole, this research investigates the conceptual configuration design for deployable space truss structures that are line-foldable with the help of graph theory. First, the bijection between a truss and its graph model is established. Therefore, operations can be performed based on graph models. Second, by introducing Maxwell’s rule, maximum clique, and chordless cycle, the principle of conceptual configuration synthesis is analyzed. A corresponding procedure is formed and it is verified by a truss with seven nodes. Third, assisted by some theorems of graph theory, the simplified double-color topological graph of deployable space truss structures is acquired and it also displays the procedure with a case. Finally, based on the above analysis, it obtains the optimal conceptual configurations. This novel research lays the foundation for kinematic synthesis and geometric dimension designs.

On the Strong Hanani-Tutte Theorem

A graph is planar if it has a drawing in which no two edges cross. The Hanani-Tutte Theorem states that a graph is planar if it has a drawing $D$ such that any two edges in $D$ cross an even number of times. A graph $G$ is a non-separating planar graph if it has a drawing $D$ such that (1) edges do not cross in $D$, and (2) for any cycle $C$ and any two vertices $u$ and $v$ that are not in $C$, $u$ and $v$ are on the same side of $C$ in $D$. Non-separating planar graphs are closed under taking minors and hence have a finite forbidden minor characterisation. In this paper, we prove a Hanani-Tutte type theorem for non-separating planar graphs. We use this theorem to prove a stronger version of the strong Hanani-Tutte Theorem for planar graphs, namely that a graph is planar if it has a drawing in which any two disjoint edges cross an even number of times or it has a chordless cycle that enables a suitable decomposition of the graph.

Growth and arrest of topological cycles in small physical networks

The chordless cycle sizes of spatially embedded networks are demonstrated to follow an exponential growth law similar to random graphs if the number of nodesNxis below a critical valueN*. For covalent polymer networks, increasing the network size, as measured by the number of cross-link nodes, beyondN*results in a crossover to a new regime in which the characteristic size of the chordless cyclesh*no longer increases. From this result, the onset and intensity of finite-size effects can be predicted from measurement ofh*in large networks. Although such information is largely inaccessible with experiments, the agreement of simulation results from molecular dynamics, Metropolis Monte Carlo, and kinetic Monte Carlo suggests the crossover is a fundamental physical feature which is insensitive to the details of the network generation. These results show random graphs as a promising model to capture structural differences in confined physical networks.

A Cycle of Maximum Order in a Graph of High Minimum Degree has a Chord

A well-known conjecture of Thomassen states that every cycle of maximum order in a $3$-connected graph contains a chord. While many partial results towards this conjecture have been obtained, the conjecture itself remains unsolved. In this paper, we prove a stronger result without a connectivity assumption for graphs of high minimum degree, which shows Thomassen's conjecture holds in that case. This result is within a constant factor of best possible. In the process of proving this, we prove a more general result showing that large minimum degree forces a large difference between the order of the largest cycle and the order of the largest chordless cycle.

Snarks with total chromatic number 5

Graph Theory International audience A k-total-coloring of G is an assignment of k colors to the edges and vertices of G, so that adjacent and incident elements have different colors. The total chromatic number of G, denoted by χT(G), is the least k for which G has a k-total-coloring. It was proved by Rosenfeld that the total chromatic number of a cubic graph is either 4 or 5. Cubic graphs with χT = 4 are said to be Type 1, and cubic graphs with χT = 5 are said to be Type 2. Snarks are cyclically 4-edge-connected cubic graphs that do not allow a 3-edge-coloring. In 2003, Cavicchioli et al. asked for a Type 2 snark with girth at least 5. As neither Type 2 cubic graphs with girth at least 5 nor Type 2 snarks are known, this is taking two steps at once, and the two requirements of being a snark and having girth at least 5 should better be treated independently. In this paper we will show that the property of being a snark can be combined with being Type 2. We will give a construction that gives Type 2 snarks for each even vertex number n≥40. We will also give the result of a computer search showing that among all Type 2 cubic graphs on up to 32 vertices, all but three contain an induced chordless cycle of length 4. These three exceptions contain triangles. The question of the existence of a Type 2 cubic graph with girth at least 5 remains open.

Graphs with no induced wheel and no induced antiwheel

A wheel is a graph that consists of a chordless cycle of length at least 4 plus a vertex with at least three neighbors on the cycle. An antiwheel is the complementary graph of a wheel. It was shown recently that detecting induced wheels is an NP-complete problem. In contrast, it is shown here that graphs that contain no wheel and no antiwheel have a very simple structure and consequently can be recognized in polynomial time.

On Efficient Coloring of Chordless Graphs

We are given a simple graph G = (V, E). Any edge e ∈ E is a chord in a path P ⊆ G (cycle C ⊆ G) iff a graph obtained by joining e to path P (cycle C) has exactly two vertices of degree 3. A class of graphs without any chord in paths (cycles) we call path-chordless (cycle-chordless). We will prove that recognizing and coloring of these graphs can be done in O(n2) and O(n) time, respectively. Our study was motivated by a wide range of applications of the graph coloring problem in coding theory, time tabling and scheduling, frequency assignment, register allocation and many other areas.

AN ORDER DEGREE ALTERNATOR FOR ARBITRARY TOPOLOGIES

An alternator is an arbitrary set of interacting processes that satisfies three conditions. First, if a process executes its critical section, then no neighbor of that process can execute its critical section at the same state. Second, along any infinite sequence of system states, each process will execute its critical section, an infinite number of times. Third, along any maximally concurrent computation, the alternator will stabilize to a sequence of states in which the processes will execute their critical sections in alternation. A principal reason for interest in alternators is their ability to transform systems correct under serial execution semantics to systems that are correct under concurrent execution semantics. An earlier alternator for arbitrary topology required 2q states where q is the dependency graph circumference and after stabilization would wait 2q steps between critical section executions. In a synchronous environment, this alternator requires only 2d+1 states where d is the degree of the graph of process dependencies for the system and after stabilization will require a wait of 2d+1 steps between critical section executions. In an asynchronous environment, the synchronization properties of this alternator must be supplemented with an asynchronous unison algorithm. The asynchronous unison algorithm requires expansion of the required number of states to dt, where t is the longest chordless cycle in the dependency graph; however, the required wait between critical section executions remains O(d).

EFFICIENT PARALLEL ALGORITHMS FOR FINDING CHORDLESS CYCLES IN GRAPHS

We present simple and efficient parallel algorithms for obtaining a chordless cycle of length greater than or equal to k in a graph whenever such a cycle exists. Our results generalize and simplify existing results for detecting chordless cycles.

Powers of chordal graphs

An undirected simple graph G is called chordal if every circle of G of length greater than 3 has a chord. For a chordal graph G, we prove the following: (i) If m is an odd positive integer, Gm is chordal. (ii) If m is an even positive integer and if Gm is not chordal, then none of the edges of any chordless cycle of Gm is an edge of Gr, r < m.