A Reconfigurations Analogue of Brooks' Theorem and Its Consequences

Abstract

Let G be a simple undirected connected graph on n vertices with maximum degree . Brooks' Theorem states that G has a proper -coloring unless G is a complete graph, or a cycle with an odd number of vertices. To recolor G is to obtain a new proper coloring by changing the color of one vertex. We show an analogue of Brooks' Theorem by proving that from any k-coloring, k > , a -coloring of G can be obtained by a sequence of O(n 2 ) recolorings using only the original k colors unless -G is a complete graph or a cycle with an odd number of vertices, or -k = + 1, G is -regular and, for each vertex v in G, no two neighbors of v are colored alike.

Toward Cereceda's conjecture for planar graphs

“…Conjecture has resisted some efforts and has only been verified (other than for trees) for graphs with degeneracy at least where denotes the maximum degree of the graph . It is also known to hold if degeneracy is replaced by tree‐width .…”

Introduction to Reconfiguration

“…The results on connectivity also showed quadratic bounds on diameter, including a lower bound on diameter for chordal graphs, culminating in the proof of the conjecture for k ≥ tw(G) + 2 [85]. Subsequently, the conjecture was proved for k = ∆(G) + 1 and ∆(G) ≥ 3; in this case, the reconfiguration graph consists of isolated vertices and one more component, with diameter O(|V(G)| 2 ) [81]. It was further observed that by increasing the number of colors, it is possible to obtain linear diameter, in particular, for every k ≥ 2col(G) + 2 [86].…”

Introduction to Reconfiguration

“…Subsequently, the conjecture was proved for k = ∆(G) + 1 and ∆(G) ≥ 3; in this case, the reconfiguration graph consists of isolated vertices and one more component, with diameter O(|V(G)| 2 ). [81]. It was further observed that by increasing by number of colors, it is possible to obtain linear diameter, in particular, for every k ≥ 2col(G) + 2 [86].…”

“…Recall that the degeneracy, col(G), is the largest minimum degree of any subgraph of G; the degeneracy of G is an upper bound on its maximum degree, and the treewidth of G, tw(G), is an upper bound on col(G). More precisely, for any connected graph that is not regular, col(G) = ∆(G) − 1 [81], and the chromatic number is at most one greater than the degeneracy. Yet another property of use is the Grundy number, χ g (G), defined as the largest possible number of colors used by a greedy coloring of G; χ g (G) ≤ ∆(G) + 1.…”

“…For bounded-degree graphs, reachability can be solved in constant time when ∆(G) ≤ k − 2, linear time when k ≥ 3 and ∆(G) = k − 1, and quadratic time when k = 3 and ∆(G) ≥ 3 [81]. With respect to parameterized complexity, the problem is fixed-parameter tractable when parameterized by k + (where is the length of the reconfiguration sequence) [79,87] but W[1]-hard when parameterized by alone [79].…”