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 lec(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{lec}\,}}(G)$$\end{document}. We determine the precise value of this parameter for any simple graph G of diameter at least 3. We show that deciding, whether lec(G)=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{lec}\,}}(G) = 2$$\end{document} for graphs G of diameter 2, is an NP-complete problem. Furthermore, we characterize all complete bipartite graphs Kr,s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{r,s}$$\end{document} with lec(Kr,s)=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{lec}\,}}(K_{r,s}) = 2$$\end{document}.


Introduction and motivation
In this paper, we consider simple and undirected graphs only.For notation and graph theoretic terminology, we generally follow the book of West [18].Aside from that, we denote by n = |V (G)| the order of a graph G and by m = |E(G)| the size of G.The degree of a vertex v of G, denoted by deg G (v), is the number of edges of G incident with the vertex v.We will denote by ∆(G) and by δ(G) the maximum degree and the minimum degree of the graph G. Furthermore, for simplicity, we denote by [k, l] the set of integers {k, k + 1, . . ., l} if k ≤ l.Note that [k, l] = ∅ for k > l.
Two edges are adjacent if they share a common vertex.Two edges are semi-adjacent if they are not adjacent but there is a third edge that is adjacent to both of them.
The weight w(e) of an edge e = uv is the sum of degrees of its end-vertices, i.e. w(e) = deg(u)+deg(v).The reduced maximum weight rw(G) of the graph G is defined as rw(G) = max{w(e) − 1 : e ∈ E(G)}.
Let G be a graph.A path L in G is a subgraph of G consisting of a vertex set {v 1 , v 2 , . . ., v k } and edge set {v 1 v 2 , v 2 v 3 , . . ., v k−1 v k }.For simplicity, we write L : v 1 v 2 . . .v k and, if L contains two vertices u, v, then we denote by uLv the subpath of L between u and v. Two vertices, say u, v ∈ V (G), are connected by a path L if the end vertices of L are u and v.Such a path is denoted as an u, v-path.
The circumference of a graph G, cir(G), is the length of the longest cycle.We will denote by C k (resp.K k , resp.K r,s ) the cycle (resp.the complete graph, resp.complete bipartite graph) on k ≥ 3 (resp.k, resp.r ≥ 1 and s ≥ 1) vertices.
Let A = {a, b, c, . ..} be a finite alphabet, i.e. a set of colours, digits, symbols, . . ., whose elements are called letters.A word A of length k over A is a sequence of letters, say A : a 1 a 2 a 3 . . .a k where a i ∈ A for all i ∈ [1, k].Let us recall some properties that words can have (see e.g.[10]).A word A is . . .proper if consecutive letters are not identical, . . .rainbow if it does not contain two identical letters, . . .conflict-free if at least one letter occurs exactly once in A, . . .monochromatic if all letters are identical, . . .odd if each letter of the alphabet A appears an odd number of times or zero times in A, . . .loose if either A consists of exactly one letter, of exactly two different letters, or has at least three different letters, Using properties of words over an alphabet A, Brause, Jendrol', and Schiermeyer [3] have introduced a graph theoretic meta-concept as follows.
Consider a graph G.If there is an edge colouring φ : E(G) → A, then we can associate to a path Now, let P be a property of words over the alphabet A. Considering an edge colouring φ : E(G) → A we say that L has the property P if the associated word φ(v 1 v 2 )φ(v 2 v 3 ) . . .φ(v k−1 v k ) has property P.
Let G be a connected graph, A be an alphabet with k letters, P be a property of words, and φ : E(G) → A be an edge-colouring.The edge-colouring φ makes G to be P edge-connected if any two distinct vertices of G are connected by a path having the property P. The minimum integer k, for which there exists an edge colouring φ : E(G) → A with |A| = k that makes G to be P edge-connected, is the P edge-connection number of G.
From a practical point of view, the P edge-connection of graphs plays an important role for security and accessibility in communication networks.While the information which one sends through a network from one node to another has to be protected by passwords, it is of high importance that the password sequences of information transferring paths meet some prescribed requirements.Since managing a whole bunch of passwords, that are assigned to direct information transferring paths between two nodes, is expensive, it is a natural question to ask for a minimum number of passwords securing the information transferring paths.By representing each node as a vertex, each direct information transferring path between two nodes as an edge, each possible password by a different colour, and the prescribed requirements for the password sequences of information transferring paths by a property P for a word over the alphabet of colours, the above problem is translated to finding the P edge-connection number of a path.
Chartrand et al. [8] introduced the concept of rainbow connection; that is, the concept of the P edge-connection where property P is rainbow.Let rc(G) denote the rainbow edge-connection number.Returning to the mathematical problem and its applications, the P edge-connection number of graphs have been intensively studied for some of the mentioned properties, for example if property P is rainbow [7,8,12,13,14,15], proper [1,2], odd [3], or conflict-free [11].In this paper, we concentrate on the loose edge-connection, which is a relaxation of the concepts rainbow connection and conflict-free connection.A motivation to study the loose edge-connection comes also from the loose-colouring of embedded graphs, see e.g.[17], [9], or [10].Note, that the loose vertex-connection of simple connected graphs is studied in a more general version in [4].
We say that a path L, which is a subgraph of a graph G with an edge colouring φ : loose edge-coloured if the path consists of one edge, or exactly two differently coloured edges, or has at least three edges that are coloured with at least three distinct colours.
A connected graph G, coloured by an edge colouring φ : E(G) → [1, k], is said to be loose edgeconnected if any two distinct vertices are connected by a loose coloured path.The least integer k, for which we have an edge colouring φ : E(G) → [1, k] that makes G loose edge-connected, is called the loose edge-connection number of G, denoted by lec(G).
Observe, that any rainbow connected graph is also loose connected.Also, if a graph G is a spanning subgraph of a graph H and lec(G) = k, then lec(H) ≤ k.
In the sequel, a path in an edge coloured graph will be called monochromatic, bi-chromatic, and loose if its edges are coloured with exactly one colour, exactly two colours, and with at least three colours, respectively.
A block B of a connected graph G is an induced subgraph of G of order at least 2 which is 2-connected and, with respect to this condition, maximal; that is, The block-cutpoint graph of a graph G is a bipartite graph B(G) in which one partite set consists of the cut-vertices of G, and the other has a vertex b i for each block B i of G.We include vb i as an edge of B(G) if and only if v ∈ B i .When G is connected, its block-cutpoint graph is a tree whose leaves are blocks of G.These blocks are called leaf blocks.If B is a trivial block and e ∈ E(B) is the edge of B, then e is a cut-edge of G; that is, G − e is disconnected.
Let C(G) be the graph induced on the set E(C) ⊆ E(G) of all cut-edges of G consisting of all vertices of G that are incident to the edges of the set E(C).More precisely, We denote by ∆(C(G)) the maximum degree of C(G).The graph C(G) is called the cut-edge graph of G. (see [18], p. 156).Let F t be a graph obtained from a graph F by attaching t leaves, t ≥ 0, to every vertex of F .Let H be a non-trivial connected graph, F be a 2-connected graph, and t ≥ 0 be an integer.A graph H is of type (t, F) if F ⊆ H ⊆ F t , and H contains a vertex adjacent with exactly t leaves.
The rest of this paper is organized as follows: Sections 2 -7 contain some preliminary statements that are necessary for the proofs of our main results.In Sections 8 the characterization of all complete bipartite graphs with lec(K r,s ) = 2 is given.In Section 10, the precise values of the loose edge-connection number for all non-trivial connected graphs of diameter at least 3 are determined .Proof.For the edge-colouring of G we use the following adaptation of the FSB algorithm (see e. g. [18], p. 99).

Loose edge-connection of trees
For a tree G let M = [1, rw(G)] be the set of colours.
Input: A tree G, a start vertex u, and the colour set M .Idea: Maintain a set R of vertices that have been reached by coloured edges but not searched and a set S of vertices that have been searched, i.e. vertices of the set S are incident only with coloured edges.The set R is maintained as a First-In First-Out list (queue), so the first vertices are the first vertices explored.
Initialization: R = {u}, S = ∅.Iteration: As long as R = ∅, we search for the first vertex v of R. Then we colour each edge e = vw of G incident with v and w / ∈ R ∪ S, with colour ϕ(e) = min{M \ F (e)}. (Here set F (e) is the set of colours used till now on edges adjacent or semi-adjacent with e.) Next the neighbour w of v is added to the back of R. If all the edges incident with v are coloured, the vertex v is removed from the front of R and is placed in S.
It is easy to see that this algorithm works properly.
3 Graphs of type R Proof.Case 1.First we consider the graph R t .Observe that diam(R t ) = 3 and lec(R t ) ≥ max{3, t}.
The following loose edge-colouring with three colours is suitable for We need to show that t + 1 colours are necessary for t ≥ 3. We start with the graph R 3 .Assume that three colours are enough to get a loose connection of R 3 .Then all three leaves attached to each vertex of C 3 of R 3 are coloured with three different colours.Consider the vertices x and y.Let, w.l.o.g., ϕ But for any j, j = i, we also have {ϕ(xz), ϕ(yz Assume that there is a loose colouring of R t with t colours for t ≥ 4, ϕ and let colours 1, 2, and 3 are used on the edges of C 3 .If we delete from R t the edges coloured with colour k ∈ [4, t], we get a loose colouring of R 3 with three colours.This contradicts the nonexistence of such a colouring of R 3 . It is easy to see that any graph H of type (t, R) which is a subgraph of G 1 has a loose edge-colouring with t colours.
We need to show that t + 1 colours are necessary for t ≥ 3 if H has two vertices, say, x and z, whose degree sum in C(H) is at least 2t−1.Assume for the contrary that the graph H has a loose edge-colouring ϕ with t colours.Consider the graph H and its vertices x, x i , i ∈ [1, t−1], z, and This completes the proof of our Lemma 2.
4 Graphs of type Q , then the following hold: A suitable loose edge-colouring of Q t , t ≥ 2, with at least four colours follows: Case 2. If t ≥ 4, then the loose colouring of H with max{3, t} colours follows from the loose colouring of Q t with t colours.Case 3. Let t = 2. Subcase 3.1 Let on C 4 of H there are at most two vertices each adjacent with exactly two leaves.Then it is a subgraph of one of the following graphs: It is easy to see that the graph H has a loose edge-colouring with three colours obtained from one of the above colouring of the graph H 1 or H 2 .
Subcase 3.2 Let on C 4 of H there are at least three vertices, say x, v, and y, each adjacent with exactly two leaves.Assume that there is a loose edge-colouring φ with three colours.We can assume, w.l.o.g., that φ(xx 1 ) = φ(vv 1 ) = a and φ(xx 2 ) = b.To the end of the proof in this case it is enough to show that the graph Q * = (Q 3 \ {yy 3 }) ∪ {xy} does not have any loose edge-colouring with three colours.Assume the opposite.Let it has such a colouring ϕ.
Let the neighbour leaf y i of y be adjacent to y via the edge yy i coloured with a colour i ∈ {1, 2}.Then the following situation, w.l.o.g., appears in Q * : {ϕ(xv), ϕ(vy)} = {1, 3} and {ϕ(xw), ϕ(wy)} = {2, 3} because there are loose x 1 , y 1 -path and x 2 , y 2 -path of length al least three.This, w.l.o.g., enforces ϕ(xy) = 1 and then no loose x 3 , v 3 -path exists.A contradiction It is easy to see that colouring ϕ of G 1 just described has the required properties of loose edge-coloring.From this colouring of G 1 one easily derives a loose edge-colouring with three colours of any subgraph , and ϕ(yy 3 ) = ϕ(ww 3 ) = 3.From this colouring of G * one easily derives a loose edge-colouring with three colours of any subgraph H of G * , which is of type (t, Q).Case 6.Let the graph H contain both diagonals xy and vw.Then H ⊆ G 2 = Q 3 ∪ {xy, vw}.It is easy to find a suitable loose edge-colouring of G 2 with three colours when we color edges xv and yw with colour 1, the edges xw and yv with colour 2, and the edges xy and vw with colour 3.
To get a suitable loose edge-colouring of P 2 we first take a subgraph P 1 of P 2 and colour its edges as above and then colour the remaining edges of P 2 as follows: ϕ(uu 2 ) = a, ϕ(yy 2 ) = b, and ϕ(xx There is no loose edge-colouring of P 3 with three colours.Assume, for a contrary, that there is one, φ, with colours a, b, and c.Observe, that there must be an edge, say, w.l.o.g., uy, with φ(uy) = b and φ(e) = b for all e ∈ {ux, vx, vz, yz}.As all three colours have to be used on the the edges incident to leaves adjacent to u as well as to y, there are two edges uu i and yy j with φ(uu i ) = φ(yy j ) = a = b.Then there is no loose u i , y j -path, a contradiction.
The following is a suitable loose colouring of P t , t ≥ 3: Here {a, b, c, d} ⊆ [1, t].This gives lec(P 3 ) = 4 and lec(P t ) = t for t ≥ 4. It is easy to see that any graph H ⊆ P t , which is of type (t, P) for t ≥ 4, has lec(H) = t.
Case 2. Let t = 3, H contain no diagonal on its cycle C 5 , and have at least two vertices adjacent to exactly three leaves.Assume that H has a loose edge-colouring φ with three colours 1, 2, and 3. Evidently, all these three colours have to appear on C 5 and let, w.l.o.g., colour 3 be unique on C 5 .
Subcase 2.1.The graph H contains on C 5 two adjacent vertices, say u and x, each of which is adjacent to exactly 3 leaves.Let φ(xx i ) = φ(uu i ) = i.Colour 3 cannot appear on the edge ux because otherwise there is no loose u 1 , x 1 -path and we would have a contradiction.Let a ∈ {1, 2}.If {φ(xv), φ(vz)} = {a, 3} then there is no loose x a , z-path in H; a contradiction.The case {φ(xu), φ(uy)} = {a, 3} is analogous.Hence, in this case, H requires four colours for its loose edge-colouring.As H ⊆ P 3 we can use the loose edge-colouring of P 3 for H. Subcase 2.2.The graph H contains on C 5 two non-adjacent vertices, say x and y, each of which is adjacent to exactly 3 leaves.Let φ(xx i ) = φ(yy i ) = i.If {φ(ux), φ(uy)} = {a, 3} then there is no loose x a , y a -path in H; a contradiction.If {φ(xv), φ(vz)} = {a, 3} then there is no loose x a , z-path in H; a contradiction.The case when {φ(yz), φ(vz)} = {a, 3} is symmetric to the latter one.This means that H requires four colours for its loose edge-colouring.Now we use the loose edge-colouring of P 3 for H.This completes the proof of our Lemma 4.
For the completeness we mention the following:

Loose edge-connection of 2-connected graphs
The complete solution for 2-connected graphs is given in the following.
For its proof we first show the following lemma.
Proof.We have to show how to colour the edges of the graph G with colours from the set A = {a, b, c} to get a loose edge-connection of G. Our procedure depends on the circumference cir(G) of G.
If cir(G) = 3, then we have to colour the graph K 3 , see Theorem 1, the complete graph on three vertices.It will play an important role later.It is easy to see that lec(K 3 ) = 1.
If cir(G) = 4, then we have to color the graphs from the set B 4 , see Theorem 1.
Let the loose edge-colouring of K 2,r be as follows: r], ϕ(yv 1 ) = b, ϕ(yv 2 ) = c and ϕ(yv i , ) = a, i ∈ [3, r].Observe, that between any two vertices of K 2,r , except for the pair x and y, there is a loose (i.e. at least three coloured) path.For the pair x and y, if r ≥ 3, there are three different bi-chromatic x, y-paths of length three using the same three colours.
A loose edge-colouring of the graph K ′ 2,r is obtained from that of K 2,r using ϕ(xy) = a.If cir(G) = 5, then we have to color the graphs from the set B 5 , see Theorem 1.Let the loose edge colouring of P r,s be defined as follows: Note that for r ≥ 2, s ≥ 1, and any pair of vertices of P r,s , except for the pair x and y, there is, in P r,s , a loose path using three colours.For the pair x and y there are three differently bi-colored x, y-paths of length two using the same three colours.The graph P ′ r,s (P ′′ r,s ), r ≥ 1, s ≥ 1, comes from the graph P r,s by inserting the edge xy (the edges xy and xz, respectively).The inserted edge in P ′ r,s is coloured with colour ϕ(xy) = b (The inserted edges in P ′′ r,s are colored with colours ϕ(xy) = c and ϕ(xz) = b).A loose edge-colouring of the graph K + 2,r is obtained from that one of K 2,r using colour a for the edge v 2 v 3 .
The proof that the loose edge-connectivity of the remaining graphs from B 5 is at most 3 is easy and is left to the reader.
If cir(G) ≥ 6, we proceed as follows.Colour the edges of the longest cycle C alternatively by 1, 2, 3, 1, 2, 3, .. and other edges of G with colour 1.We distinguish three cases.Case 1..For any two vertices x and y on the cycle C there is a path of length at least three, which is a loose path.
Case 2. If x ∈ V (C) and y / ∈ V (C), then there are two paths connecting y with two vertices z 1 , z 2 on the cycle C.At least one of them is distinct from x, say z 1 .By Case 1 there is a path L on C of length at least 3 from z 1 to x, which is loose (uses three colors).Now, taking the path from z 1 to y (which is vertex disjoint with L except of vertex z 1 ) we obtain a loose x, y-path.
Case 3. Let x, y / ∈ V (C).Similarly as above, there are two vertex disjoint paths from x to z 1 and from y to z 2 with z 1 , z 2 ∈ V (C), y = z 2 .As in Case 2 there is a loose path from z 1 to z 2 along the cycle C which leads to a loose path from x to y.
Corollary 1.For every two vertices x and y of a 2-connected graph G with cir(G) ≥ 6 there is a loose x, y-path.
Proof.It follows directly from the proof of Lemma 6.
Proof.Theorem 2. To prove it first observe the following statements.
1. lec(G) = 1 if and only if rc(G) = 1 if and only if G is complete graph.

lec(G) = 2 if and only if rc
Now for all remaining 2-connected graphs we have This proves the theorem.

Loose edge-connection number of complete bipartite graphs
The complete solution for the complete bipartite graphs is given in the following.
Theorem 3. If K r,s , r ≥ s ≥ 1, is a complete bipartite graph, then the following statements hold: (iii) lec(K r,s ) = 3 otherwise.
Proof.Let K r,s be a complete bipartite graph with the vertex set V (K r,s ) = X∪Y , where X = {x 1 , . . ., x r } and Y = {y 1 , . . ., y s }.Let S i = K 1,s be the star with the vertex set First we consider case (ii).Suppose that φ is an edge-colouring of K r,s with two colours a and b.Recall that a path in an edgecoloured graph is called bi-chromatic (resp.monochromatic) if on its edges exactly two colors (resp.one) are (is) used.
Let s(i) = (a i,1 , . . ., a i,s ) be a vector of colours of the edges of the star S i such that φ(x i y j ) = a i,j for all i ∈ [1, r] and j ∈ [1, s].
This colouring has the following obvious two properties: Observe, that for r > 2 s there is no loose edge-coloring of K r,s with two colours.The reason is that the number of different binary vectors of dimension s is 2 s .Then, by the pigeonhole principle, there is a pair of stars, S k and S l , with the same vector of colours, s(k) and s(l), respectively, and, by Claim 1, we have no bi-chromatic path of length two between the pair of vertices x k and x l .
If 2 ≤ s ≤ r ≤ 2 s , then there exists a loose edge-colouring of K r,s .It is enough to consider the vectors of colours for the stars S i , i ∈ [1, s − 1], with the following properties: φ(x i y j ) = a for j ≤ i and φ(x i y j ) = b for i + 1 ≤ j ≤ n.For i ∈ [s, r] we colour the edges of stars S i with such vectors of colours that are mutually distinct from already chosen ones.
To see that this edge-colouring is loose, observe that Claim 1 ensures us that for any pair of vertices x k and x l , 1 ≤ k < l ≤ r, there is a bi-chromatic path of length two between them.The choice of the first s − 1 vectors of colour ensures the existence of a bi-chromatic y k , y l -path of length two for any pair of vertices y k and y l for any pair k and l with 1 ≤ k < l ≤ n.See Claim 2.
This completes the proof of the theorem for the case (ii).
The proof of theorem in the case (i) follows from Lemma 1 and in the case (iii) from Lemma 6.

Graphs with diameter at most 2
To continue we need some more definitions.
The join of two simple graphs G and H, written G ∨ H, is the graph obtained from the disjoint union G + H by adding the edges {xy : x ∈ V (G), y ∈ V (H)}, see [18].
A nontrivial block B of a graph G is called large if cir(B) ≥ 6 and it is called small if 2 ≤ cir(B) ≤ 5.
Recall that a block B is called trivial if it consists of a cut-edge e.In this case we sometimes will write B = e.
It is easy to see that graphs of diam(G) ≥ 3 have the loose edge-connection number lec(G) ≥ 3.So the necessary condition for graphs G to have lec(G) = 2 is to have diam(G) = 2.In the previous section we have characterized all complete bipartite graphs B that have lec(B) = 2.
Observe, that if a graph H is a spanning subgraph of a non-complete graph G and lec(H) = 2, then lec(G) = 2 Let G and H be graphs with loose edge-connection number two.Then it is easy to prove that lec(G ∨ H) = 2, and lec(K In Theorem 2 we have mentioned that for a graph G it holds lec(G) = 2 if and only if rc(G) = 2. Several classes of graphs G with rc(G) = 2 are known, see e.g.[5], [17], or [16].For example, Caro et al. [5] have proved that any non-complete n-vertex graph with minimum degree It is known (cf.[6]) that deciding whether rc(G) = 2 is an NP-complete problem.Hence, the problem to decide whether lec(G) = 2 is also an NP-complete.
The following problem seems to be interesting.Proof.Part 1. Colouring.
5 , K 5 } be a set of specific small blocks.Recall that any large block and any small block, other than that from D, has a loose edge-colouring with three colours.Between any two distinct vertices of large blocks and any two vertices of small blocks other than that from D, except two specific x and y of small blocks, there exists a loose path (Recall, that a path is loose if has length at least three, which edges are coloured with at least three colours).The vertices x and y are connected with three mutually distinct bi-chromatic x, y-path.See the proof of Lemma 6.
Our colouring follows the properties of the block-cutpoint graph B(G) of G. Choose a vertex of B(G) corresponding to a block B as a root.We colour edges of the blocks from this root to the leaf blocks.First we colour edges of B as described in Lemma 6 if B is 2-connected and with colour 1 if B is trivial.If B is trivial (i.e.B = {e}, where e is an edge) we colour all edges of the component of C(G) containing the edge e according to the method used in the proof of Lemma 1.
General Step: Let v be a cut vertex on B and let B 1 , . . ., B k , S 1 , . . ., S l , T 1 , . . ., T r be other blocks containing v as cut-vertex, where B i , for i ∈ [1, k], is a large block, S j , for j ∈ [1, l], is a small block, and T s , for s ∈ [1, r], is a trivial block.First we colour large blocks, next small blocks different from blocks from D, then small blocks from D, and, finally, trivial blocks.
We distinguish three cases.
Case 1.Let deg B (v) = 1 and ϕ(wv) = a be a colour of the unique edge wv in B. Then for every i ∈ [1, k] the edges of the block B i are coloured with colours from the color set {a, b, c} as stated in Corollary 1.
If S j , j ∈ [1, l], is a small block from (B 3 ∪ B 4 ∪ B 5 ) \ D and deg S j (v) = 2, then the edges incident to v in S j are colored with colours from {a, b, c} so that the vertex v becomes of type (b, c), where b = a = c.Next we finish the colouring of edges of S j as described in the proof of Lemma 6.If deg S j (v) ≥ 3, then we colour S j directly as in Lemma 6.
If S j ∈ D, then its edges are coloured together with all trivial blocks (edges) incident to the vertices of the block S j .In this case we colour the subgraph H of type (t, O) for t ≥ 1 and O ∈ {P, Q, R} as described in Lemmas 4, 3, and 2, respectively, taking into consideration the following three facts: (i) The edges of the longest cycle of H incident with the vertex v have to obtain in H the colours b and c. (ii) Some of edges of H at v can already be coloured (This can happen if the block B is trivial, or there are several blocks of types from {P, Q, R} at the considered cut vertex v).(iii) If H is of type (1, R), we colour its edges with three colours as in R 1 because H ⊂ R 1 .
If S j ∈ D and there no trivial blocks incident to vertices of S j , i.e. S j is subgraph of type (0, O), we colour its edges as in Lemma 6 except of the case when S j = C 3 = K 3 .In the later case the edges of C 3 are coloured with three distinct colours.
Let there be trivial blocks at v, i.e., r ≥ 1.If this block is a part of a subgraph of type O ∈ {P, Q, R} at v, then all trivial blocks are coloured as a part of the colouring of the first considered subgraph of type O.If there is no subgraph of type O ∈ {P, Q, R} at v, then the edges of all trivial blocks (i.e.edges) are part of a component of C(G).In this case we colour all edges of this component as described in the proof of Lemma 1. Case 3. Let deg B (v) ≥ 3. Our colouring in this case continues as in Case 2 keeping in mind that all small blocks S j , j ∈ [1, l], have to be colored in such a way that the cut-vertex v becomes of type (a, b) if it is of degree two in S j .
If all blocks containing v as a cut-vertex are colored, v is called ready.
To continue in the colouring of G a next, not ready, cut-vertex from a colored block is chosen.The procedure of colouring finishes if every cut vertex of G is ready.Observe, see Lemmas 1, 2, 3, 4 and 6, that our colouring uses colours from the set M = [1, rw(C(G))] except for the cases listed in our Theorem 5.These exceptional cases use the colours from the colour set M * = [1,4] if G contains a subgraph H of type Q described in Lemma 3 (i) and from the colour set M ′ = [1, rw(C(G)) + 1] if G contains subgraphs H of type O ∈ {P, Q, R} described in the Lemmas 4 (ii), 3 (ii), and 2 (iii), respectively.Part 2. Loose connectivity.To finish the proof of the theorem it is necessary to show that any two distinct vertices w 1 and w 2 are connected by an edge w 1 w 2 , or bi-chromatic w 1 , w 2 -path of length 2, or a loose (with at least three different colours) w 1 , w 2 -path of length at least 3.Such a path exists if both w 1 and w 2 belong to the same block, to the same component of C(G), or to the same subgraph H of type O ∈ {P, Q, R}.
If there is an w 1 , w 2 -path going through two vertices, say u 1 and u 2 , of a large block or of a small block, then the part between them can be replaced by a loose u 1 , u 2 -path lying inside the block.
Hence, it is sufficient to show the existence of such a loose w 1 , w 2 -path for any two blocks D 1 and D 2 with a common cut-vertex z, w 1 ∈ V (D 1 ) and w 2 ∈ V (D 2 ), w 1 = z = w 2 .
If the block D 1 is large, then there is a loose w 1 , z-path in D 1 , which together with a z, w 2 -path in D 2 gives a needed loose w 1 , w 2 -path.
Let the block D 1 be small and D 2 be either small or trivial.If the vertex w 1 is not corresponding to any of the vertices x and y of D 1 ∈ {K 2,s , K ′ 2,s , K + 2,s } or to any of the vertices x, y and z of D 1 ∈ {P 2,s , P ′ 2,s , P ′′ 2,s }, respectively, then there is, in D 1 , a loose w 1 , z-path.This path together with a z, w 2path gives a required loose w 1 , w 2 -path.
If the vertex w 1 and the vertex w 2 are corresponding to some of the above mentioned vertices x, y, or z of D 1 and of D 2 , respectively, Then one can easy recognise in G a loose w 1 , w 2 -path of length 3 or 4.

Lemma 1 .
If G is a tree of the reduced weight rw(G), then lec(G) = rw(G).

Case 3 .
, and ϕ(zz t ) = t.Let ϕ(xz) = c.Then the edges xy and yz have to be coloured with different colours, say a and b.Then there is neither a loose x a , z a -path nor a loose x b , z b -path in H.A contradiction.Let t ≤ 2. If H contains at least three leaves, then diam(H) = 3, H ⊆ R 2 , and, by Case 1, lec(H) = 3.If H has exactly two leaves, then one can easily show that lec(H) = 3 and an edge-colouring of C 3 consists of three different colours.If H has one leaf, we colour all the edges of the 3-cycle of C 3 with colour 1 and remaining edges of Hwith colour 2.

Case 2 . 3 .
Let H contain on C 5 exactly one vertex, say v, adjacent to exactly three leaves.Then H is subgraph of the graph G 3 = P 2 ∪ {v 3 } ∪ {vv 3 }, which has a loose edge-colouring obtained from the colouring of P 2 extended by φ(vv 3 ) = a.This colouring of G 3 induces a loose edge-colouring of H with three colours.Case 3. Let H contain the cycle C 5 with a diagonal.Then, w.l.o.g., H contains a subgraph G 4 = (V (G 4 ), E(G 4 )) with vertex set V (G 4 ) = {x, y, z, u, v} and the edge set E(G 4 ) = {ux, vx, uy, yz, xy, vz}.If we colour the edges of E(G 4 ) as follows φ(vx) = φ(xy) = 1, φ(ux) = φ(yz) = 2, and φ(uy) = φ(vz) = 3, we can easily extend this colouring to a loose edge-colouring of H with three colours.Case 4. If H contains at least two diagonals on its C 5 , then we solve first the situation for its subgraph with one diagonal as in Case 3, and then colour arbitrarily the remaining diagonals.

Lemma 5 .Subcase 2 .
If G t is a graph obtained from the cycle C k , k ≥ 6, by attaching t leaves, t ≥ 1, to every vertex of C k , then lec(G t ) = max{3, t}.replace the 3-path x v vwx w with the edge x v x w .The resulting graph B contains a 3-cycle and B ∈ B 5 or B contains a cycle of length at least 6, a contradiction.Let the minimum degree δ(B) ≥ 3. Consider a vertex v ∈ V (B) with deg B (v) = δ(B).Then the subgraph B − v is connected and each non-trivial block of it is in B 3 ∪ B 4 ∪ B 5 .One can easily see that then B contains a cycle of length at least 6, a contradiction.

Case 2 .
Let deg B (v) = 2 and the vertex v be of type (a, b).Then the large block B i , for any i ∈ [1, k] is coloured with colours from the color set {a, b, c} as in the proof of Lemma 6.The block S j , for any j ∈[1, l], S j / ∈ D,is handled as a large block if v has, in it, degree at least three, or such that the vertex v becomes of type (a, b) in the colouring of S j , if v has degree two.If S j ∈ D is a subgraph H of type O ∈ {P, Q, R}, then we consider the subgraph H of type O and colour it as described in Case 1. Trivial blocks T s are handled in the same way as in Case 1.

if and only if H contains two vertices b 1 and b 2 such that deg C(H)
vv 1 , vy, yy 1 , yy 2 , yw, ww 1 , wx} and a loose edgecolouring φ(xx 1