Francis Guthrie's approach to The Four Color Problem
The odd wheel is the only type of 4-critical graph in which one vertex always gets a unique color. This supports Frederic Guthrie's approach to the Four Color Problem.
math.GM↗
arXiv subjects
Asbjørn Brændeland
Publications and source records attributed to Asbjørn Brændeland.
Split-by-edges trees
A split-by-edges tree of a graph G on n vertices is a binary tree T where the root = V(G), every leaf is an independent set in G, and for every other node N in T with children L and R there is a pair of vertices {u, v} in N such that L = N - v, R = N - u, and uv is an edge in G. It follows from the definition that every maximal independent set in G is a leaf in T, and the maximum independent sets of G are the leaves closest to the root of T.
cs.DS↗
A family of greedy algorithms for finding maximum independent sets
The greedy algorithm A iterates over a set of uniformly sized independent sets of a given graph G and checks for each set S which non-neighbor of S, if any, is best suited to be added to S, until no more suitable non-neighbors are found for any of the sets. The algorithms receives as arguments the graph, the heuristic used to evaluate the independent set candidates, and the initial cardinality of the independent sets, and returns the final set of independent sets.
cs.DS↗
Depth-first search in split-by-edges trees
A layerwise search in a split-by-edges tree (as defined by Brændeland, 2015) of agiven graph produces a maximum independent set in exponential time. A depth-first search produces an independent set, which may or may not be a maximum, in linear time, but the worst case success rate is maybe not high enough to make it really interesting. What may make depth-first searching in split-by-edges trees interesting, though, is the pronounced oscillation of its success rate along the graph size axis.
cs.DS↗
Rule based lexicographical permutation sequences
In a permutation sequence built by means of sub permutations the transition between successive permutations are subject to a set of n(n - 1)/2 rules that group into n - 1 matrices with a high degree of regularity. By means of these rules the sequence can be produced in O(3n!) time and O(n^3) space.
cs.DS↗
nCk sequences and their difference sequences
A nCk sequence is a sequence of n-bit numbers with k bits set. Given such a sequence C, the difference sequence D of C is subject to certain regularities that make it possible to generate D in 2|C| time, and, hence, to generate C in 3|C| time.
cs.DS↗
Color fixation and color identity in 4-chromatic graphs
I argue that there is no 4-chromatic planar graph with a joinable pair of color identical vertices, i.e., given a 4-chromatic planar graph G and a pair of vertices {u, v} in G, if the color of u equals the color of v in every 4-coloring of G, then there is no planar supergraph of G where u and v are adjacent. This is equivalent to the Four Color Theorem. (My argument is a variation of my argument in arXiv:1402.7368)
math.GM↗
Color identical pairs in 4-chromatic graphs
I argue that, given vertices u and v in a 4-chromatic graph G, if the color of u equals the color of v in every 4-coloring of G then G has no planar supergraph where u and v are adjacent. This is equivalent to the Four Color Theorem.
math.GM↗