Resumen de Contribucions a la teoria de l'aresta-acoloriment de grafs: snarks i multipols
Joan Vilaltella Castanyer
A graph where every vertex has three neighboring vertices is a cubic graph. An edge-coloring is an assignment of colors to the edges of a graph in such a way that the edges incident to a vertex have no repeated colors. An edge-coloring is optimal if it uses the minimum possible number of colors. Vizing's Theorem implies that an optimal edge-coloring of a cubic graph requires three or four colors. If three colors are enough, we call the edge-coloring a Tait-coloring. If four colors are needed, we call the graph a snark. Holyer proved that deciding wether a cubic graph is Tait-colorable is an NP-complete problem, therefore it is widely believed that it is a very difficult or intractable problem in the general case. Nevertheless, theory does not forbid efficient solutions in specific cases: this is called "breaking intractability". We describe an heuristic algorithm called CVD, for "Conflicting Vertex Displacement", which has a good empirical performance in random regular graphs. Also it allows us to check a conjecture by Biggs on "odd graphs" in instances with milions of vertices and edges, using moderately powerful computers. Snarks are relevant in graph theory: they appear often as minimal counterexamples of important conjectures, such as the Cycle Double Cover Conjecture (every bridgeless graph has a family of cycles such that every edge belongs to exactly two cycles of the family). In the analysis and synthesis of snarks, multipoles, "pieces" of cubic graphs with free ends that can be joined to each other, are often used. In a Tait-colored multipole, the number of equally colored free ends for each color and the total number of free ends have the same parity (the number of vertices of the multipole has this same parity, too). This result, known as the Parity Lemma, allows the interpretation of multipoles as logic gates and cubic graphs as logic circuits. This gives a very general way to construct snarks, based on logic circuits with no valid Boolean assignment, and allows us to relate the Tait-coloring of cubic graphs to integer factorization. In particular, we can construct snarks from prime numbers. A state of a multipole is the restriction of a Tait-coloring of the multipole to its free ends. If the set of states of a multipole is a non-empty subset of the set of states of another multipole with a larger number of vertices, we call the smaller multipole a reduction of the larger one. An irreducible multipole is a multipole with no reduction (an obvious example is a minimal multipole, that is, with no vertices or with a single vertex). The maximum number of vertices of an irreducible multipole as a function of its number (m) of free ends is denoted by v(m). Its behavior is well-known only for m<6, while there is a specific lower bound for v(6). We prove the irreducibility of multipoles having a tree, a forest or a cycle as their underlying graphs. This allows us to prove a linear lower bound for v(m), the first general result for this function.
