Edge-colored Graph Research Articles

Graphs with large (1,2)-rainbow connection numbers

Let G be an edge-colored graph. If every subpath of length at most l+1 within a path P in a graph G consists of uniquely colored edges, then P is called an l-rainbow path. A connected graph G is deemed (1,2)-rainbow connected if there exists at least one 2-rainbow path connecting two distinct vertices within G. The minimum number of colors needed to attain (1,2)-rainbow connectedness in a connected graph G, represented as rc1,2(G), is referred to as the (1,2)-rainbow connection number. If G is a nontrivial connected graph of size m, then rc1,2(G)=m if and only if G is the star or double star of size m. Our main goal is to identify all connected graphs G of size m that satisfy the condition m−3≤rc1,2(G)≤m−1.

Rainbow Cycles in Properly Edge-Colored Graphs

We prove that every properly edge-colored n-vertex graph with average degree at least 32(log5n)2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$32(\\log 5n)^2$$\\end{document} contains a rainbow cycle, improving upon the (logn)2+o(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\log n)^{2+o(1)}$$\\end{document} bound due to Tomon. We also prove that every properly edge-colored n-vertex graph with at least 105k3n1+1/k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^5 k^3 n^{1+1/k}$$\\end{document} edges contains a rainbow 2k-cycle, which improves the previous bound 2ck2n1+1/k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{ck^2}n^{1+1/k}$$\\end{document} obtained by Janzer. Our method using homomorphism inequalities and a lopsided regularization lemma also provides a simple way to prove the Erdős–Simonovits supersaturation theorem for even cycles, which may be of independent interest.

Rainbow disjoint union of P4 and a matching in complete graphs

An edge-colored graph G is called rainbow if all the colors on its edges are distinct. Given graphs G and H, the anti-Ramsey number AR(G,H) is the maximum number k such that there exists an edge-coloring of G with exactly k colors containing no rainbow copy of H. Recently, the anti-Ramsey problem for disjoint union of graphs received much attention. In particular, several researchers focused on the problem for graphs consisting of small components. In this paper, we continue the work in this line. We refine the bound of n and determine the precise value of AR(Kn,P4∪tP2) for all n≥2t+4 in complete graphs.

Rainbow spanning trees in random subgraphs of dense regular graphs

We consider the following random model for edge-colored graphs. A graph G on n vertices is fixed, and a random subgraph Gp is chosen by letting each edge of G remain independently with probability p. Then, each edge of Gp is colored uniformly at random from the set [n−1]. A result of Frieze and McKay (Random Structures and Algorithms, 1994) implies that if ε>0, G=Kn, and p=(2+ε)log⁡nn, then Gp almost surely contains a rainbow spanning tree. In this paper, we show that if ε>0 and G is a d-regular Ω(n)-edge-connected graph, then when p=(2+ε)log⁡nd, Gp almost surely contains a rainbow spanning tree. Our main tool is a new edge-replacement method for rainbow forests.

Edge-disjoint properly colored cycles in edge-colored complete graphs

In an edge-colored graph G, let dmon(v) denote the maximum number of edges with the same color incident with a vertex v in G, called the monochromatic-degree of v. The maximum value of dmon(v) over all vertices v∈V(G) is called the maximum monochromatic-degree of G, denoted by Δmon(G). Li et al. in 2019 conjectured that every edge-colored complete graph G of order n with Δmon(G)≤n−3k+1 contains k vertex-disjoint properly colored (PC for short) cycles of length at most 4, and they showed that the conjecture holds for k=2. Han et al. showed that every edge-colored complete graph G of order n with Δmon(G)≤n−2k contains k PC cycles of different lengths. They further got the condition Δmon(G)≤n−6 for the existence of two vertex-disjoint PC cycles of different lengths. In this paper, we consider the problems of the existence of edge-disjoint PC cycles of length at most 4 (different lengths) in an edge-colored complete graph G of order n.

Color-avoiding percolation and branching processes

AbstractWe study a variant of the color-avoiding percolation model introduced by Krause et al., namely we investigate the color-avoiding bond percolation setup on (not necessarily properly) edge-colored Erdős–Rényi random graphs. We say that two vertices are color-avoiding connected in an edge-colored graph if, after the removal of the edges of any color, they are in the same component in the remaining graph. The color-avoiding connected components of an edge-colored graph are maximal sets of vertices such that any two of them are color-avoiding connected. We consider the fraction of vertices contained in color-avoiding connected components of a given size, as well as the fraction of vertices contained in the giant color-avoidin g connected component. It is known that these quantities converge, and the limits can be expressed in terms of probabilities associated to edge-colored branching process trees. We provide explicit formulas for the limit of the fraction of vertices contained in the giant color-avoiding connected component, and we give a simpler asymptotic expression for it in the barely supercritical regime. In addition, in the two-colored case we also provide explicit formulas for the limit of the fraction of vertices contained in color-avoiding connected components of a given size.

Compatible Spanning Circuits and Forbidden Induced Subgraphs

A compatible spanning circuit in an edge-colored graph G (not necessarily properly) is defined as a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. The existence of extremal compatible spanning circuits (i.e., compatible Hamilton cycles and compatible Euler tours) has been studied extensively. Recently, sufficient conditions for the existence of compatible spanning circuits visiting each vertex at least a specified number of times in specific edge-colored graphs satisfying certain degree conditions have been established. In this paper, we continue the research on sufficient conditions for the existence of such compatible s-panning circuits. We consider edge-colored graphs containing no certain forbidden induced subgraphs. As applications, we also consider the existence of such compatible spanning circuits in edge-colored graphs G with κ(G) ≥ α(G), κ(G) ≥ α(G) − 1 and κ (G) ≥ α(G), respectively. In this context, κ(G), α(G) and κ (G) denote the connectivity, the independence number and the edge connectivity of a graph G, respectively.

On the Turán number of the hypercube

Abstract In 1964, Erdős proposed the problem of estimating the Turán number of the d-dimensional hypercube $Q_d$ . Since $Q_d$ is a bipartite graph with maximum degree d, it follows from results of Füredi and Alon, Krivelevich, Sudakov that $\mathrm {ex}(n,Q_d)=O_d(n^{2-1/d})$ . A recent general result of Sudakov and Tomon implies the slightly stronger bound $\mathrm {ex}(n,Q_d)=o(n^{2-1/d})$ . We obtain the first power-improvement for this old problem by showing that $\mathrm {ex}(n,Q_d)=O_d\left (n^{2-\frac {1}{d-1}+\frac {1}{(d-1)2^{d-1}}}\right )$ . This answers a question of Liu. Moreover, our techniques give a power improvement for a larger class of graphs than cubes. We use a similar method to prove that any n-vertex, properly edge-coloured graph without a rainbow cycle has at most $O(n(\log n)^2)$ edges, improving the previous best bound of $n(\log n)^{2+o(1)}$ by Tomon. Furthermore, we show that any properly edge-coloured n-vertex graph with $\omega (n\log n)$ edges contains a cycle which is almost rainbow: that is, almost all edges in it have a unique colour. This latter result is tight.

Chromatic Polynomials of 2-Edge-Coloured Graphs


 
 
 Using the definition of colouring of $2$-edge-coloured graphs derived from 2-edge-coloured graph homomorphism, we extend the definition of chromatic polynomial to 2-edge-coloured graphs. We find closed forms for the first three coefficients of this polynomial that generalize the known results for the chromatic polynomial of a graph. We classify those graphs that admit a 2-edge-colouring for which the chromatic polynomial of the graph and the chromatic polynomial of the 2-edge-colouring is equal. Finally, we examine the behaviour of the roots of this polynomial, highlighting behaviours not seen in chromatic polynomials of graphs.
 
 

Properly colored cycles in edge-colored 2-colored-triangle-free complete graphs

An edge-colored graph Gc is called properly colored if any two adjacent edges receive distinct colors. An edge-colored graph Gc is 2-colored-triangle-free if Gc contains no 2-colored-triangle, where a 2-colored-triangle is an edge-colored triangle with exactly two colors. Let dc(v) be the number of colors on the edges incident to v in Gc and let δc(Gc) be the minimum dc(v) for all v∈V(Gc). In this paper we extend the definitions of proper vertex-pancyclic and proper edge-pancyclic to proper k-path-pancyclic, defined as follows: An edge-colored graph Gc is said to be proper k-path-pancyclic if each properly colored path of length k is contained in a properly colored cycle of length l for every l with max{3,k+2}≤l≤|V(Gc)|. We prove that an edge-colored 2-colored-triangle-free complete graph Gc with δc(Gc)≥k+3 is either (almost) proper k-path-pancyclic or contains a large monochromatic complete subgraph.

Anti-Ramsey number of matchings in outerplanar graphs

A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. Given a graph family G and a graph H, the anti-Ramsey number AR(G,H) is the maximum number of colors in an edge-coloring of a graph G∈G which has no rainbow copy of H. When G={G}, we write AR(G,H) for short. Erdős, Simonovits and Sós introduced these numbers in the 1970s, they studied the case when G is a complete graph. Since then, many results have appeared considering G and H belonging to specific graph classes. In particular, Jendrol’, Schiermeyer and Tu (Jendrol’ et al., 2014) obtained an upper bound for AR(Tn,kK2) in terms of n and k for k≥3 and n≥2k, where Tn is the family of maximal planar graphs on n vertices. Jin and Ye (Jin and Ye, 2018) continued the study of anti-Ramsey numbers of matchings in planar graphs, by deriving an upper bound for AR(Mn,kK2), where Mn is the family of maximal outerplanar graphs on n vertices. In both works, examples are shown for which there are colorings using many colors while avoiding rainbow matchings, but the numbers of colors in these examples do not reach the upper bounds. The bounds of AR(Tn,kK2) were improved in following works by other researchers. This paper goes in the same direction of these two works, proving exact value for AR(Bn,kK2) in terms of n and k for n≥6 and k≥3, where Bn is the family of maximal bipartite outerplanar graphs on n vertices which contains a Hamilton cycle.

Edge-colouring graphs with local list sizes

The famous List Colouring Conjecture from the 1970s states that for every graph G the chromatic index of G is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph G with sufficiently large maximum degree Δ and minimum degree δ≥ln25⁡Δ, the following holds: for every assignment L of lists of colours to the edges of G, such that |L(e)|≥(1+o(1))⋅max⁡{deg⁡(u),deg⁡(v)} for each edge e=uv, there is an L-edge-colouring of G. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, k-uniform hypergraphs, and recently Molloy generalized Kahn's original result to correspondence colouring as well as its hypergraph generalization. We prove local versions of all of these generalizations by showing a weighted version that simultaneously implies all of our results.

Tiling Edge-Coloured Graphs with Few Monochromatic Bounded-Degree Graphs

Tiling Edge-Coloured Graphs with Few Monochromatic Bounded-Degree Graphs

Finding Large Rainbow Trees in Colourings of $K_{n,n}$

A subgraph of an edge-coloured graph is called rainbow if all of its edges have distinct colours. An edge-colouring is called locally $k$-bounded if each vertex is incident with at most $k$ edges of the same colour. Recently, Montgomery, Pokrovskiy and Sudakov showed that for large $n$, a certain locally 2-bounded edge-colouring of the complete graph $K_{2n+1}$ contains a rainbow copy of any tree with $n$ edges, thereby resolving a long-standing conjecture by Ringel: For large $n$, $K_{2n+1}$ can be decomposed into copies of any tree with $n$ edges. In this paper, we employ their methods to show that any locally $k$-bounded edge-colouring of the complete bipartite graph $K_{n,n}$ contains a rainbow copy of any tree $T$ with $(1- o(1))n/k$ edges. We show that this implies that every tree with $n$ edges packs at least $n$ times into $K_{n+o(1),n+o(1)}$. We conjecture that for large $n$, $K_{n,n}$ can be decomposed into $n$ copies of any tree with $n$ edges.

Local Irregularity Conjecture vs. cacti

A graph in which the two end-vertices of every edge have distinct degrees is called locally irregular. An edge coloring of a graph G such that every color induces a locally irregular subgraph of G is said to be locally irregular. A graph G which admits a locally irregular edge coloring is called colorable. The Local Irregularity Conjecture claims that 3 colors are enough for locally irregular edge coloring of any colorable graph. Not so long ago the only known graphs which do not admit a locally irregular 3-edge coloring were the non-colorable graphs, all of which are cacti. This fact motivated us to study locally irregular 3-edge colorings of cacti, which first resulted with the finding of the bow-tie graph B, a colorable cactus graph which requires 4 colors for a locally irregular edge coloring. As the graph B disproves the stated form of the Local Irregularity Conjecture, here we continue the study of locally irregular 3-edge colorings of cacti in relation to the conjecture. In this paper we confirm that all other cacti are locally irregular 3-edge colorable. This result makes us to believe that B is the only counterexample to the conjecture not only in the class of cacti, but also in general.

Trianguloids and triangulations of root polytopes

Triangulations of a product of two simplices and, more generally, of root polytopes are closely related to Gelfand-Kapranov-Zelevinsky's theory of discriminants, to tropical geometry, tropical oriented matroids, and to generalized permutohedra. We introduce a new approach to these objects, identifying a triangulation of a root polytope with a certain bijection between lattice points of two generalized permutohedra. In order to study such bijections, we define trianguloids as edge-colored graphs satisfying simple local axioms. We prove that trianguloids are in bijection with triangulations of root polytopes.

Improved bounds for the triangle case of Aharoni's rainbow generalization of the Caccetta-Häggkvist conjecture

For a digraph G and v∈V(G), let δ+(v) be the number of out-neighbors of v in G. The Caccetta-Häggkvist conjecture states that for all k≥1, if G is a digraph with n=|V(G)| such that δ+(v)≥k for all v∈V(G), then G contains a directed cycle of length at most ⌈n/k⌉. Aharoni proposed a generalization of this conjecture, that a simple edge-colored graph on n vertices with n color classes, each of size at least k, has a rainbow cycle of length at most ⌈n/k⌉. Let us call (α,β)triangular if every simple edge-colored graph on n vertices with at least αn color classes, each with at least βn edges, has a rainbow triangle. Aharoni, Holzman, and DeVos showed the following:•(9/8,1/3) is triangular;•(1,2/5) is triangular. In this paper, we improve those bounds, showing the following:•(1.1077,1/3) is triangular;•(1,0.3988) is triangular. Our methods give results for infinitely many pairs (α,β), including β<1/3; we show that (1.3481,1/4) is triangular.

Euler dynamic H-trails in edge-colored graphs

Alternating Euler trails has been extensively studied for its diverse practical and theoretical applications. Let H be a graph possibly with loops and G be a multigraph without loops. In this paper we deal with any fixed coloration of E(G) with V(H) (H-coloring of G). A sequence W = ( v 0 , e 0 1 , … , e 0 k 0 , v 1 , e 1 1 , … , e n − 1 k n − 1 , v n ) in G, where for each i ∈ { 0 , … , n − 1 } , k i ≥ 1 and e i j = v i v i + 1 is an edge in G, for every j ∈ { 1 , … , k i } , is a dynamic H-trail if W does not repeat edges and c ( e i k i ) c ( e i + 1 1 ) is an edge in H, for each i ∈ { 0 , … , n − 2 } . In particular, a dynamic H-trail is an alternating trail when H is a complete graph without loops and ki = 1, for every i ∈ { 1 , … , n − 1 } . In this paper, we introduce the concept of dynamic H-trail, which arises in a natural way in the modeling of many practical problems, in particular, in theoretical computer science. We provide necessary and sufficient conditions for the existence of closed Euler dynamic H-trail in H-colored multigraphs. Also we provide polynomial time algorithms that allows us to convert a cycle in an auxiliary graph, L 2 H ( G ) , in a closed dynamic H-trail in G, and vice versa, where L 2 H ( G ) is a non-colored simple graph obtained from G in a polynomial time.

Properly colored cycles of different lengths in edge-colored complete graphs

A cycle in an edge-colored graph is called properly colored if all of its adjacent edges have distinct colors. Let Knc be an edge-colored complete graph with n vertices and let k be a positive integer. Denote by Δmon(Knc) the maximum number of edges of the same color incident with a vertex of Knc. In this paper, we show that (i) if Δmon(Knc)⩽n−2k, then Knc contains k properly colored cycles of different lengths and the bound is sharp; (ii) if Δmon(Knc)⩽n−2k+1−2k+4, then Knc contains k vertex-disjoint properly colored cycles of different lengths; in particular, Δmon(Knc)⩽n−6 suffices for the existence of two vertex-disjoint properly colored cycles of different lengths.

Loose Edge-Connection of Graphs

In the last years, connection concepts such as rainbow connection and proper connection appeared in graph theory and obtained a lot of attention. In this paper, we investigate the loose edge-connection of graphs. A connected edge-coloured graph G is loose edge-connected if between any two of its vertices there is a path of length one, or a bi-coloured path of length two, or a path of length at least three with at least three colours used on its edges. The minimum number of colours, used in a loose edge-colouring of G, is called the loose edge-connection number and denoted {{,textrm{lec},}}(G). We determine the precise value of this parameter for any simple graph G of diameter at least 3. We show that deciding, whether {{,textrm{lec},}}(G) = 2 for graphs G of diameter 2, is an NP-complete problem. Furthermore, we characterize all complete bipartite graphs K_{r,s} with {{,textrm{lec},}}(K_{r,s}) = 2.