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### On constructing snakes in powers of complete graphs

Jerzy Wojciechowski
1998 Discrete Mathematics
chords) in the cartesian product K~ of d copies of the complete graph Kin.  ...  We prove the conjecture of Abbott and Katchalski that for every m ~> 2 there is a positive constant 2,. such that S(K~n ) >~ 2mnd-lS(Ka~ -1) where S(Ka~) is the length of the longest snake (cycle without  ...  Hence, by Lemma 2.2(ii), (rg [] 9) [] 7n = cg [] 7n q is a snake in Gp+q, and the proof is complete. []3. Construction of long snakesAssume that d >~4.  ...

### Further results on snakes in powers of complete graphs

H.L. Abbott, M. Katchalski
1991 Discrete Mathematics
Katchalski, Further results on snakes in powers of complete graphs, Discrete Mathematics 91 (1991) 111-120.  ...  Introduction By a snake in a finite graph G is meant a simple cycle without chords. We denote by S(G) the length of a longest snake in G.  ...  Let K,, denote the complete graph with n vertices and let K," denote the product of d copies of K,,.  ...

### The Minimum Shared Edges Problem on Grid-like Graphs [article]

Till Fluschnik, Meike Hatzel, Steffen Härtlein, Hendrik Molter, and Henning Seidler
2017 arXiv   pre-print
We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route p paths from a start vertex to a target vertex in a given graph while using at most k edges more  ...  We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number p of paths.  ...  This is possible since the number of vertices in G is a power of two.  ...

### The Minimum Shared Edges Problem on Grid-Like Graphs [chapter]

Till Fluschnik, Meike Hatzel, Steffen Härtlein, Hendrik Molter, Henning Seidler
2017 Lecture Notes in Computer Science
We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route p paths from a start vertex to a target vertex in a given graph while using at most k edges more  ...  We show that MSE can be decided on bounded grids in linear time when both dimensions are either small or large compared to the number p of paths.  ...  This is possible since the number of vertices in G is a power of two.  ...

### Finding Longest Paths in Hypercubes: 11 New Lower Bounds for Snakes, Coils, and Symmetrical Coils [chapter]

Seth J. Meyerson, Thomas E. Drapela, William E. Whiteside, Walter D. Potter
2015 Lecture Notes in Computer Science
INDEX WORDS: stochastic beam search; snake-in-the-box; combinatorial optimization; graph search; hypercube; heuristic search FINDING LONGEST PATHS IN HYPERCUBES: 11 NEW LOWER BOUNDS FOR SNAKES, COILS,  ...  Stochastic Beam Search, a non-deterministic variant of Beam Search, provides the overall structure for our search, while graph theory based techniques are used in the computation of a generational fitness  ...  ACKNOWLEDGMENTS We thank Dustin Cline and Thomas Horton for their important contributions to this project, Snake-in-the-Box research, and the Institute for Artificial Intelligence.  ...

### Page 7184 of Mathematical Reviews Vol. , Issue 95m [page]

1995 Mathematical Reviews
A snake in a graph is a cycle without chords. Denote by S(K/) the length of a longest snake in the product of d copies of the complete graph on n vertices.  ...  Abbott (3-AB; Edmonton, AB) 95m:05149 05C38 05C35 Wojciechowski, Jerzy (1-WV; Morgantown, WV) Long snakes in powers of the complete graph with an odd number of vertices. (English summary) J.  ...

### New Bounds for the Snake-in-the-Box Problem [article]

David Allison, Daniel Paulusma
2016 arXiv   pre-print
The Snake-in-the-Box problem is that of finding a longest induced path in an n-dimensional hypercube. We prove new lower bounds for the values n∈{11,12,13}.  ...  The Coil-in-the-Box problem is that of finding a longest induced cycle in an n-dimensional hypercube. We prove new lower bounds for the values n∈{12,13}.  ...  Acknowledgements This work made use of the facilities of the Hamilton HPC Service of Durham University.  ...

### Page 5549 of Mathematical Reviews Vol. , Issue 95i [page]

1995 Mathematical Reviews
Here a(H) is the maximum size of a snake containing no edge of H. This is analogous to the maximum size of a stable set in a random graph.  ...  We take £(H) to be the minimum number of edges of H contained in any n-snake.  ...

### The Snake Optimizer for Learning Quantum Processor Control Parameters [article]

Paul V. Klimov, Julian Kelly, John M. Martinis, Hartmut Neven
2020 arXiv   pre-print
In this whitepaper, we introduce the Snake Optimizer for efficiently and quickly solving such optimization problems by leveraging concepts in artificial intelligence, dynamic programming, and graph optimization  ...  This application enabled state-of-the-art system performance on a 53 qubit quantum processor, serving as a key component of demonstrating quantum supremacy.  ...  CONTRIBUTIONS P.V.K. conceived the Snake Optimizer. P.V.K and J.K. developed ideas for calibration stack integration. J.M.M. and H.N. supported research and development. V.  ...

### The firefighter problem for graphs of maximum degree three

Stephen Finbow, Andrew King, Gary MacGillivray, Romeo Rizzi
2007 Discrete Mathematics
We show that the firefighter problem is NP-complete for trees of maximum degree three, but in P for graphs of maximum degree three if the fire breaks out at a vertex of degree at most two.  ...  These lead to bounds on the maximum number of vertices that can be saved. NP-completeness of the firefighter problem on bipartite graphs is established in  .  ...  In the construction of (T , r), these two clauses gave rise to 12 snake trees: Claim 2.  ...

### Lengths of snakes in boxes

Ludwig Danzer, Victor Klee
1967 Journal of Combinatorial Theory
A snake in a graph G is a simple circuit C in G such that C has no chords in G--that is, every edge of G which joins two vertices of C is in fact an edge of C.  ...  For each d ~ 2 let S(d) denote the length (number of vertices) of the longest snake in the graph I a formed by the vertices and edges of the d-dimensional cube.  ...  Then one of the graphs P C~ I s and P c5 1, 8 has at least five vertices and thus is as in one of the last two diagrams above.  ...

### Page 23 of Mathematical Reviews Vol. 55, Issue 1 [page]

1978 Mathematical Reviews
Snakes in powers of complete graphs. SIAM J. Appl. Math. 32 (1977), no. 2, 347-355.  ...  In this paper, some snakes are constructed for the class of graphs K,° (product of d copies of the complete graph K,) and thus some lower bounds for the length of the longest snakes are obtained.  ...

### Long Snakes in Powers of the Complete Graph with an Odd Number of Vertices

Jerzy Wojciechowski
1994 Journal of the London Mathematical Society
In  Abbott and Katchalski ask if there exists a constant c > 0 such that for every d ≥ 2 there is a snake (cycle without chords) of length at least c3 d in the product of d copies of the complete graph  ...  product of d copies of the complete graph K n .  ...  Hence by Property 2 (ii ), γ n ⊗ (D ⊗ C) is a snake in K n × K d n × H = K d+1 n × H, and the proof is complete. Construction of Well Distributed Chains Let C be a chain of paths in a graph G.  ...

### Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings [article]

Ilke Canakci, Ralf Schiffler
2015 arXiv   pre-print
The definition of the rings requires the snake graph calculus which is completed in this paper building on two earlier articles on the subject.  ...  One of the main results of the current paper is the completion of the explicit construction of these bijections.  ...  The construction of abstract snake graphs and band graphs is completely detached from triangulated surfaces. Our goal is to study these objects in a combinatorial way.  ...

### The Parameterized Complexity of Motion Planning for Snake-Like Robots [article]

Siddharth Gupta, Guy Sa'ar, Meirav Zehavi
2019 arXiv   pre-print
Unfortunately, already on grid graphs, this problem is PSPACE-complete [Biasi and Ophelders, 2016].  ...  Nevertheless, we prove that even on general graphs, the problem is solvable in time k^O(k)|I|^O(1) where k is the size of the snake, and |I| is the input size.  ...  Roughly speaking, given t instances of Hamiltonian Cycle on grid graphs, our construction of an instance of Snake Game is as follows.  ...
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