Properly colored subgraphs in edge-colored graphs

Abstract

1.1.Terminology and notations 5 edge set E(X , Y ), respectively.Let C be a cycle, i.e., a (sub)graph consisting of distinct vertices v 1 , v 2 , . . ., v k and edges v i v i+1 for i = 1, 2, . . ., k -1 and v k v 1 .We can consider a fixed direction on C, either oriented from v 1 to v 2 , etc., or oriented in the opposite direction.If we have fixed one of these directions, then for two vertices u, v ∈ V (C), we use uC v to denote the segment between u and v along the direction of C, and use u C v to denote the segment between u and v along the opposite direction of C. In particular, ifAn edge-coloring of G is defined as a mapping C : E(G) → , where is the set of natural numbers.If G has such an edge-coloring, then G is an edgecolored graph.We say that a cycle C is properly colored (or simply PC) if all of its adjacent edges have distinct colors, and we say that a cycle C is rainbow if all of its edges have distinct colors.A properly colored cycle-factor of an edge-colored graph G is a spanning subgraph of G such that each component is a properly colored cycle, i.e., a set of vertex-disjoint properly colored cycles covering the vertex set.An edge-colored graph G is properly colored (even) vertex-pancyclic if every vertex of G is contained in properly colored cycles of all possible (even) lengths.Let G be an edge-colored graph.We use C(G) and c(G) to denote the set and the number of colors appearing on the edges of G, respectively.If c(G) = k, then G is called k-edge-colored.For nonempty sets X , Y ⊆ V (G), let C(X ) denote the set of colors appearing on the edges with both end-vertices in X , and C(X , Y ) denote the set of colors appearing on the edges with one end-vertex in X and the other vertex in Y .For simplicity, we use, is defined as the number of colors appearing on the edges incident with v and with other end-vertices in V (H); the i-colored degree of v to H, denoted by d i H (v), is the number of edges of color i incident with v and with other end-vertices in V (H); and the maximum monochromatic degree of v to H, denoted by ∆ mon H (v), is the maximum number of edges of the same color incident with v and with other end-vertices in H.Note that it is possible that v ∈ V (H).Problem 1.1.Given an edge-colored graph G c , check whether G c contains a PC cycle.Yeo [90] in 1997 characterized edge-colored graphs containing no PC cycles.Theorem 1.1 (Yeo [90]).If G c contains no PC cycles, then G c contains a vertex v such that no components of G cv are joined to v with edges of more than one color.Note that one can delete such vertices one by one without destroying any PC cycle.It is easy to see that Problem 1.1 is polynomially solvable. Fujita et al.[41] in 2018 obtained a sharp color degree condition for the existence of PC cycles.The problem of giving sufficient conditions for the existence of specific PC cycles seems more difficult.It was studied extensively by various researchers in the last decades. Short PC cyclesFirstly we focus on short PC cycles in general edge-colored graphs.We start by showing the Caccetta-Häggkvist Conjecture [24], which is one of the best known in graph theory.Conjecture 1.1 (Caccetta and Häggkvist [24]).For every positive integer r, every digraph on n vertices with minimum outdegree at least n/r has a directed cycle of length at most r.This conjecture is trivial for r 2 but for r 3 it remains open.Seymour and Spirkl [82] in 2020 considered a bipartite version of the Caccetta-Häggkvist Conjecture.See [85] for more partial results.Aharoni et al. [3] in 2019 considered a generalized version of the Caccetta-Häggkvist Conjecture, and obtained some results for the existence of rainbow triangles. Chapter 2 Edge-colored complete graphs containing no PC odd cyclesIn this chapter, we characterize edge-colored complete graphs containing no PC odd cycles and give an efficient algorithm with complexity O(n 3 ) for deciding the existence of PC odd cycles in an edge-colored complete graph of order n.Moreover, we show that for two integers k, m with m k 3, where k -1 and m are relatively prime, an edge-colored complete graph contains a PC cycle of length ≡ k (mod m) if and only if it contains a PC cycle of length ≡ k (mod m), where < 2m 2 (k -1) + 3m.Let x be a vertex in X 1 such that C(x, X 1 ) = {c 3 } and x a vertex inn -3 that C(x , X 1 ) = {c 3 }.This implies that there exists an edge x x with x ∈ X 1 \ x such that C(x x ) = c 3 .Since C(x, X 1 ) = {c 3 }, we have C(x x ) = c 3 .It follows that v x x x y 2 v is a PC 5-cycle, a contradiction.Case 3. X 1 = and X 2 = .Note that Y 1 = U 1 = and Y 2 = U 2 = .If |C(G)| 2, then G contains no PC odd cycles.Since ∆ mon (G) n -3, it follows from Theorem 4.5 that G contains a PC cycle of length at least 4, a contradiction.So |C(G)| 3. Let H be the subgraph induced by all edges with colors different from c 1 and c 2 .If H contains two nonadjacent edges y 1 y 2 and y 1 y 2 with y 1 , y 1 ∈ Y 1 and y 2 , y 2 ∈ Y 2 , then y 1 y 2 y 2 y 1 y 1 is a PC 4-cycle, a contradiction.So H is a star.Let u be the center of star H and let G = Gu.Then ∆ mon (G ) n -3 = |V (G )| -2, which implies that |C(G )| = 2.Note that G contains no PC odd cycles.By Theorem 4.5, G contains a PC cycle of length at least 4, which is also a PC cycle of G, a contradiction.Lemma 4.3.Let C = x yzw x be a PC 4-cycle in K c n , where C(x y) = C(zw) = c 1 and C( yz) = C(x w) = c 2 .If uv is an edge vertex-disjoint with C such that 60Chapter 4. PC cycles of different lengths in edge-colored complete graphs C(uv) = c 1 and d c 2 C (u) + d c 2 C (v) 5, then K c n [V (C) ∪ {u, v}] contains a PC 6-cycle.Proof.Suppose to the contrary that K c n [V (C) ∪ {u, v}] contains no PC 6cycles.Assume w.l.o.g. that d c 2 C (u) d c 2 C (v).Note that 3 d c 2 C (u) 4 since d c 2 C (u) + d c 2 C (v) 5.If d c 2 C (u) = 3, then assume that C(u, {x, y, z}) = {c 2 }.Since ux yzw vu, u y x wz vu and uzw x y vu are not PC 6-cycles, we have C(v, { y, z, w}) = {c 1 }, which implies d c 2 C (v) 1.If d c 2 C (u) = 4, then C(u, V (C)) = {c 2 }.Since ux yzw vu, u y x wz vu,uzw x y vu and uwz y x vu are not PC 6-cycles, we have C(v, V (C)) = {c 1 }, which implies d c 2 C (v) = 0.In both cases, d c 2 C (u) + d c 2 C (v) 4, a contradiction.