D-resolvability of vertices in planar graphs

It is well known that a vertex of degree 5 in a planar graph is not Dreducible. However, it remains an open question whether all vertices in such a graph areD-resolvable. If so, it would be a stronger result than mere 4-colorability. To investigate the question, we designed and implemented two algorithms, one to 4-color a planar graph and the other to generate internally 6-connected triangulations randomly. The coloring algorithm greedily obtains a k-coloring and then, if k > 4 (highly likely), transforms the k-coloring into a 4-coloring solely by means of Kempe exchanges (generally possible only if all vertices are D-resolvable). We used the random generator to test the coloring algorithm systematically in over 200,000 trials that included many tens of thousands of non-isomorphic graphs. We also tested the algorithm on the graphs of Errera, Kittell, Heawood and others. We encountered no failures in any of the tests. Submitted: March 2017 Accepted: April 2017 Final: April 2017 Published: June 2017 Article type: Regular paper Communicated by: G. Liotta E-mail address: [email protected] (James A. Tilley) 650 James A. Tilley D-resolvability of vertices in planar graphs Terminology and background In this article, we consider vertex-colorings of planar graphs and use the integers 1, 2, 3, 4, . . . to indicate color labels for the vertices. We define a proper coloring of such a graph G to be a partition of its vertex set V (G) into independent sets, each set assigned a distinct color. We consider proper colorings only and refer to them merely as colorings. An independent set of vertices in G is a set in which no two vertices are adjacent. If there are k such sets in a particular partition, that partition represents a k-coloring of G and, by convention, it uses consecutive color labels 1, . . . , k. Two colorings of G are considered identical if they partition V (G) into identical independent sets. Thus, two colorings of G are not considered distinct if they differ merely by a permutation of colors. We refer to a coloring of G as a state of G. In a given state of G, a j-k Kempe chain (j 6= k) is a maximal connected subgraph induced by vertices colored either j or k. There can be several distinct, mutually disconnected j-k Kempe chains. A j-k Kempe exchange (j 6= k) in a coloring of G is the interchange of colors j and k on a non-empty proper subset of all j-k chains in G. If an i-j Kempe chain and a j-k Kempe chain (i 6= k) intersect at one or more vertices colored j, then the two chains are said to be entangled. Our coloring algorithm relies on the concept of a state transformation graph. We generalize the definition from that used in [8] by admitting the possibility of colorings of a planar graph G in which a subset V≤4 of V (G) uses colors in the color set {1,2,3,4} while its complement V>4 uses colors in the color set {5,6, . . . }. The state transformation graph HG associated with G is a graph of all distinct states of G in which the colors of vertices in V>4 are fixed, but the colors of vertices in V≤4 are not. Each vertex in HG represents such a state. Two states are considered adjacent if a Kempe exchange involving a pair of colors from the color set {1,2,3,4} transforms one into the other. Because Kempe exchanges induce an equivalence relation on the states of G, each component of HG is an equivalence class under Kempe exchanges and all states in a component of HG are said to be Kempe-equivalent. Let G be a planar graph and Gv the graph obtained from G by deleting an arbitrary vertex v and all its incident edges. Let R be the ring of vertices (cycle) adjacent to v in G. We distinguish two types of state in HGv : (i) initial state: R uses all colors in the color set {1,2,3,4} and (ii) solution state: R uses only two or three colors from the color set {1,2,3,4}. These terms are logical: for G to be 4-colorable, we must be able to replace v in Gv with color 1, 2, 3, or 4 not used in R; thus, given an initial state of HGv , we must be able to transform it into a solution state. If a component of HGv consists only of initial states, then each state in that component is said to be difficult because it cannot be transformed by a sequence of Kempe exchanges into a solution state. The conjecture that the coloring algorithm we present in section 2 works for all planar graphs is equivalent to the statement that, for any such G and any vertex v in G, the state transformation graph HGv has no difficult initial states. By analogy to the notion of D-reducibility (see [15]), we say that a vertex v in a planar graph G is D-resolvable if any 4-coloring of Gv can be transformed, JGAA, 21(4) 649–661 (2017) 651 as necessary, by Kempe exchanges alone, into a 4-coloring of G. To prove Dreducibility, we are constrained to resolve a deleted vertex v by interchanging colors only on Kempe chains that include at least one vertex in the ring R adjacent to v, whereas, in the coloring algorithm we present in section 2, we are permitted to interchange colors on chains that involve only vertices not in R, perhaps at a considerable remove from R. D-reducibility is independent of the structure of G exterior to R whereas D-resolvability might not be. In a sense, it is a question of “local” versus “non-local” effects. That is why a vertex of degree 5 turns out to be D-resolvable even though it is not D-reducible. But it is a one-way street: D-reducibility implies D-resolvability, not vice versa. Vertices of degree 3 and 4, trivially D-reducible, are D-resolvable. The conjecture that the state transformation graph HGv has no difficult initial states is equivalent to the statement: Every vertex in a planar graph is D-resolvable. This statement is stronger than the 4-color theorem; the conjecture clearly implies the 4-color theorem, but the converse is not true.