Arc-colored Digraph Research Articles

Disproof of a conjecture on the rainbow triangles in arc-colored digraphs

Let D be a digraph. The arc number a(D) of D is defined as the number of arcs of D. An arc-coloring of D is a mapping C: A(D)→N, where N is the set of natural numbers. The color number c(D) of D is defined as the number of colors assigned to the arcs of D. A rainbow triangle in D is a directed triangle in which every pair of arcs has distinct colors. Recently, Li et al. (2022) proved that if a(D)+c(D)=n(n−1)+⌊n24⌋+2 for an arc-colored digraph D of order n=3,4 without containing rainbow triangles, then D≅Kn↔. Thus, they conjectured that if a(D)+c(D)=n(n−1)+⌊n24⌋+1 for an arc-colored digraph D of order n≥5 without containing rainbow triangles, then D≅Kn↔. In this note, we disprove the conjecture by constructing a family of counterexamples. In fact, we show that there is not a function f(n) such that D≅Kn↔ if a(D)+c(D)=f(n) for an arc-colored digraph D of order n≥5 without containing rainbow triangles.

Rainbow triangles in arc-colored digraphs

Let D be an arc-colored digraph. The arc number a(D) of D is defined as the number of arcs of D. The color number c(D) of D is defined as the number of colors assigned to the arcs of D. A rainbow triangle in D is a directed triangle in which every pair of arcs has distinct colors. Let f(D) be the smallest integer such that if c(D)≥f(D), then D contains a rainbow triangle. In this paper we obtain f(K↔n) and f(Tn), where K↔n is a complete digraph of order n and Tn is a strongly connected tournament of order n. Moreover we characterize the arc-colored complete digraph K↔n with c(K↔n)=f(K↔n)−1 and containing no rainbow triangles. We also prove that an arc-colored digraph D on n vertices contains a rainbow triangle when a(D)+c(D)≥a(K↔n)+f(K↔n), which is a directed extension of the undirected case.

Rainbow Triangles in Arc-Colored Tournaments

Let \(T_{n}\) be an arc-colored tournament of order n. The maximum monochromatic indegree \(\varDelta ^{-mon}(T_{n})\) (resp. outdegree \(\varDelta ^{+mon}(T_{n})\)) of \(T_{n}\) is the maximum number of in-arcs (resp. out-arcs) of a same color incident to a vertex of \(T_{n}\). The irregularity \(i(T_{n})\) of \(T_{n}\) is the maximum difference between the indegree and outdegree of a vertex of \(T_{n}\). A subdigraph H of an arc-colored digraph D is called rainbow if each pair of arcs in H have distinct colors. In this paper, we show that each vertex v in an arc-colored tournament \(T_{n}\) with \(\varDelta ^{-mon}(T_n)\le \varDelta ^{+mon}(T_n)\) is contained in at least \(\frac{\delta (v)(n-\delta (v)-i(T_n))}{2}-[\varDelta ^{-mon}(T_{n})(n-1)+\varDelta ^{+mon}(T_{n})d^+(v)]\) rainbow triangles, where \(\delta (v)=\min \{d^+(v), d^-(v)\}\). We also give some maximum monochromatic degree conditions for \(T_{n}\) to contain rainbow triangles, and to contain rainbow triangles passing through a given vertex. Finally, we present some examples showing that some of the conditions in our results are best possible.

Digraphs with proper connection number two

A directed path in a digraph is proper if any two consecutive arcs on the path have distinct colors. An arc-colored digraph D is proper connected if for any two distinct vertices x and y of D, there are both proper (x,y)-directed paths and proper (y,x)-directed paths in D. The proper connection number pc→(D) of a digraph D is the minimum number of colors that can be used to make D proper connected. Obviously, if a digraph has a proper connection number, it must be strongly connected, and pc→(D)=1 if and only if D is complete. Magnant et al. showed that pc→(D)≤3 for all strong digraphs D, and Ducoffe et al. proved that deciding whether a given digraph has proper connection number at most two is NP-complete. In this paper, we give a few classes of strong digraphs with proper connection number two, and from our proofs one can construct an optimal arc-coloring for a digraph of order n in time O(n3).

Proper connection and proper-walk connection of digraphs

An arc-colored digraph D is properly (properly-walk) connected if, for any ordered pair of vertices (u,v), the digraph D contains a directed path (a directed walk) from u to v such that arcs adjacent on that path (on that walk) have distinct colors. The proper connection number pc→(D) (the proper-walk connection number wc→(D)) of a digraph D is the minimum number of colours to make D properly connected (properly-walk connected). We prove that pc→(Cn(S))≤2 for every circulant digraph Cn(S) with S⊆{1,…,n−1},|S|≥2 and 1∈S. Furthermore, we give some sufficient conditions for a Hamiltonian digraph D to satisfy pc→(D)=wc→(D)=2.

Kernels by rainbow paths in arc-colored tournaments

For an arc-colored digraph D, define its kernel by rainbow paths to be a set S of vertices such that (i) no two vertices of S are connected by a rainbow path in D, and (ii) every vertex outside S can reach S by a rainbow path in D. In this paper, we first show that it is NP-complete to decide whether an arc-colored tournament has a kernel by rainbow paths, where a tournament is an orientation of a complete graph. In addition, we show that every arc-colored n-vertex tournament with all its strongly connected k-vertex subtournaments, 3⩽k⩽n, colored with at least k−1 colors has a kernel by rainbow paths.

Note on directed proper connection number of a random graph

For an arc-colored digraph D, we say D is properly strongly connected, if for any ordered pair of vertices (x, y), D contains a directed path from x to y such that any adjacent arcs in that path have distinct colors. The directed proper connection number pc→(D) of a digraph D, is the minimum number of colors to make D properly strongly connected. Let D(n, p) denote the random digraph model, in which every arc of a digraph is chosen with probability p independently from other arcs. We prove that if p={logn+loglogn+λ(n)}/n, then with high probability, pc→(D(n,p))=2, where λ(n) tends to infinite.

(Strong) Proper Connection in Some Digraphs

An arc-colored digraph D is proper connected if any pair of vertices vi, vj e V(D) there is a proper vi - vj path whose adjacent arcs have different colors and a proper vj - vi path whose adjacent arcs have different colors. The proper connection number of a digraph D is the minimum number of colors needed to make D proper connected, denoted by spc(D). An arc-colored digraph D is strong proper connected if any pair of vertices vi, vj e V(D) there is a proper vi - vj geodesic and a proper vj - vi geodesic. The strong proper connection number of D is the minimum number of colors required to color the arcs of D in order to make D strong proper connected, denoted by spc(D). In this paper, we will show some results on spc(D) and spc(D), mostly for the case of the (strong) proper connection numbers of cacti and circulant digraphs.

Designing deterministic polynomial-space algorithms by color-coding multivariate polynomials

We introduce an enhancement of color coding to design deterministic polynomial-space parameterized algorithms. Our approach aims at reducing the number of random choices by exploiting the special structure of a solution. Using our approach, we derive polynomial-space O⁎(3.86k)-time (exponential-space O⁎(3.41k)-time) deterministic algorithm for k-Internal Out-Branching, improving upon the previously fastest exponential-space O⁎(5.14k)-time algorithm for this problem. (The notation O⁎ hides polynomial factors.) We also design polynomial-space O⁎((2e)k+o(k))-time (exponential-space O⁎(4.32k)-time) deterministic algorithms for k-ColorfulOut-Branching on arc-colored digraphs and k-Colorful Perfect Matching on planar edge-colored graphs. In k-Colorful Out-Branching, given an arc-colored digraph D, decide whether D has an out-branching with arcs of at least k colors. k-Colorful Perfect Matching is defined similarly. To obtain our polynomial-space algorithms, we show that (n,k,αk)-splitters (α⩾1) and in particular (n,k)-perfect hash families can be enumerated one by one with polynomial delay using polynomial space.

Kernels by properly colored paths in arc-colored digraphs

A kernel by properly colored paths of an arc-colored digraph D is a set S of vertices of D such that (i) no two vertices of S are connected by a properly colored directed path in D, and (ii) every vertex outside S can reach S by a properly colored directed path in D. In this paper, we conjecture that every arc-colored digraph with all cycles properly colored has such a kernel and verify the conjecture for digraphs with no intersecting cycles, semi-complete digraphs and bipartite tournaments, respectively. Moreover, weaker conditions for the latter two classes of digraphs are given.

Formula omitted]-kernels by walks in [formula omitted]-colored digraphs and the color-class digraph

Let H be a digraph possibly with loops and D a finite digraph without loops whose arcs are colored with the vertices of H (D is an H-colored digraph). V(D) and A(D) will denote the sets of vertices and arcs of D respectively. For an arc (z1,z2) of D we will denote by cD(z1,z2) its color. A directed walk (respectively directed path) (v1, v2,…,vn) in D is an H-walk (respectively H-path) if and only if (cD(v1,v2), cD(v2,v3),…,cD(vn−1,vn)) is a directed walk in H. A set K⊆V(D) is an H-kernel by walks (respectively H-kernel) if for every pair of different vertices in K there is no H-walk (respectively H-path) between them, and for every vertex u∈V(D)∖K there exists v∈K such that there exists an H-walk (respectively H-path) from u to v in D.Let D be an arc-colored digraph. The color-class digraph of D, denoted by CC(D), is defined as follows: the vertices of the color-class digraph are the colors represented in the arcs of D and (i,j) ∈ A(CC(D)) if and only if there exist two arcs namely (u,v) ∈ A(D) colored i and (v,w) ∈ A(D) colored j. In this paper we relate the concepts discussed above, the color-class digraph and the H-coloration of D, in order to prove the existence of an H-kernel by walks (respectively H-kernel).

γ-cycles in arc-colored digraphs

γ-cycles in arc-colored digraphs

Restricted domination in arc-colored digraphs

Let H = (V(H), A(H)) be a digraph possibly with loops and D = (V(D), A(D)) a digraph whose arcs are colored with the vertices of H (this is what we call an H-colored digraph); i.e. there exists a function c: A(D) → V(H); for an arc of D, f = (u, v) ∊ A(D), we call c(f) = c(u, v) the color of f. A directed walk (directed path) P = (u0, u1,…, un) in D will be called an H-walk (H-path) whenever (c(u0, u1), c(u1, u2),…, c(un-2, un-1), c(un-1, un)) is a directed walk (directed path) in H. We introduce the concept of H-kernel N, as a generalization of the two properties that define a kernel (Recall that a kernel N of a digraph D is a set of vertices N ⊆ V (D) which is independent and for each x ∊ V(D) — N, there exists an xN-arc in D). A set N ⊆ V(D) is called H-independent whenever for every two different vertices x, y ∊ N there is no H-path between them, and N is called H-absorbent whenever for each x ∊ V (D) — N there exists a vertex y ∊ N and an xy-H-path in D. The set N ⊆ V(D) will be called H-kernel if and only if it is H-independent and H-absorbent.This new concept generalizes the concepts of kernel, kernel by monochromatic paths and kernel by alternating paths.In this paper we show sufficient conditions for an infinite digraph to have an H-kernel.

Computational experiments for the problem of Hamiltonian path with fixed number of color repetitions

In this paper we consider an approach to solve the problem of Hamiltonian path with fixed number of color repetitions for arc-colored digraphs. Our approach is based on usage of local search algorithms to solve a logical model for the problem.

Complexity of trails, paths and circuits in arc-colored digraphs

We deal with different algorithmic questions regarding properly arc-colored s–t trails, paths and circuits in arc-colored digraphs. Given an arc-colored digraph Dc with c≥2 colors, we show that the problem of determining the maximum number of arc disjoint properly arc-colored s–t trails can be solved in polynomial time. Surprisingly, we prove that the determination of a properly arc-colored s–t path is NP-complete even for planar digraphs containing no properly arc-colored circuits and c=Ω(n), where n denotes the number of vertices in Dc. If the digraph is an arc-colored tournament, we show that deciding whether it contains a properly arc-colored circuit passing through a given vertex x (resp., properly arc-colored Hamiltonian s–t path) is NP-complete for c≥2. As a consequence, we solve a weak version of an open problem posed in Gutin et al. (1998) [23], whose objective is to determine whether a 2-arc-colored tournament contains a properly arc-colored circuit.

Monochromatic cycles and monochromatic paths in arc-colored digraphs

Monochromatic cycles and monochromatic paths in arc-colored digraphs

Reachability problems in edge-colored digraphs

Let D be a digraph and let G be a multidigraph whose arcs are colored with the vertices of D. A walk, W, in G is a D- walk if the consecutive colors encountered on W also form a walk in D. A set S ⊆ V ( G ) is a D- sink if any x ∈ V ( G ) - S reaches some y ∈ S on a D-walk. We say S is D- independent (resp. independent) if no two vertices of S have a D-walk (resp. have an arc) between them. Let D ∈ B 2 (resp. D ∈ B 3 ) if any finite D arc-colored digraph G always has an independent (resp. D-independent) D-sink, and let D ∈ B 1 if any finite D arc-colored tournament always has a 1-point D-sink. Sands et al. [On monochromatic paths in edge-colored digraphs, J. Combin. Theory Ser. B 33 (1982) 271–275.] showed that if D has vertex set { red , blue } and arcs ( red , red ) , ( blue , blue ) then D ∈ B 3 , a D-walk then being just a monochromatic walk. Here we give a classification of B 2 , and make inroads in the classification of B 3 and B 1 , as well we prove the strict containment B 3 ⊊ B 2 ⊊ B 1 . Finally, we generalize these problems by considering D to be an automaton and replacing D by the language accepted by D.