c-Critical Graphs with Maximum Degree Three

∗

M.A. Fiol Departament de Matem`atica Aplicada i Telem`atica Universitat Polit`ecnica de Catalunya

Abstract Let G be a (simple) graph with maximum degree three and chromatic index four. A 3-edge-coloring of G is a coloring of its edges in which only three colors are used. Then a vertex is conflicting when some edges incident to it have the same color. The minimum possible number of conflicting vertices that a 3edge-coloring of G can have, d(G), is called the edge-coloring degree of G. Here we are mainly interested in the structure of a graph G with given edge-coloring degree and, in particular, when G is c-critical, that is d(G) = c ≥ 1 and d(G − e) < c for any edge e of G.

1

Edge-colorings

Throughout this paper G = (V, E) denotes a connected (simple) graph with maximum degree ∆ = 3, order p = p(G) = | V (G) | and size q = q(G) = | E(G) |. The degree of a vertex v ∈ V will be denoted by δ(v). For i = 1, 2, 3 let pi = pi (G) denote the number of vertices of G with degree i. Then p1 + 2p2 + 3p3 = 2q and, therefore, p1 ≡ p3 mod 2. A k-edge-coloring (in wide sense) of G, denoted ΨG or Ψ, is simply an assignment of one of k colors to each of its edges. If the edges incident to each vertex have different color we will say that such a ∗

Research supported in part by the Comisi´ on Interministerial de Ciencia y Tecnologia, Spain, under grant TIC90-0712.

1

coloring is proper. The chromatic index of G, χ′ (G), is the minimum integer k such that G has a proper k-edge-coloring. As it is well-known, in our case χ′ (G) must be either 3 or 4; see Johnson [11] or Vizing [13]. In the former case G is said to be of class one. Otherwise, G is said to be of class two. The following characterization of the ”coloring” of a set of edges was used in [2] and [4] to construct snarks. Let G be a graph with a 3-edge-coloring Ψ, and consider a subset of m edges F ⊂ E, mi of which have color i, i = 1, 2, 3. Then we say that the set F has Boole coloring 0, denoted by Ψ(F ) = 0, iff m1 ≡ m2 ≡ m3 ≡ m mod 2. On the other hand, we say that F has Boole coloring 1 (or, more specifically, 1a ), denoted by Ψ(F ) = 1(1a ), iff ma + 1 ≡ mb ≡ mc ≡ m + 1 mod 2 where the letters a, b, c stand for the colors 1, 2, 3 in any order. The Boole coloring of any vertex v ∈ V , denoted by Ψ(v), is then defined to be the Boole coloring of the edges incident to it. It is interesting to note the following facts: when δ(v) = 1, Ψ(v) = 1i iff the edge incident to v has color i, i = 1, 2, 3; when δ(v) = 2, Ψ(v) = 0 (respec. Ψ(v) = 1) iff the two edges incident to v have the same (respec. different) color; when δ(v) = 3, Ψ(v) = 0 iff the three edges incident to v have different colors. It turns out that the set of Boole colorings, {0, 11 , 12 , 13 }, together with the sum operation defined in the natural way, is isomorphic to the Klein group (with 0 as the identity element, 1i + 1i = 0, and 1a + 1b = 1c ), see [4]. Note that, as each element coincides with m)

its inverse, m1i = 1i + · · · +1i is 0 when m is even and 1i when m is odd. From this fact, we have the following useful result which can be seen as a generalization of the so-called parity lemma [8]. Lemma 1.1. Let G = (V, E) be a graph with a given 3-edge-coloring, such that, for i = 1, 2, 3, ni of its vertices have Boole coloring 1i . Then, n1 ≡ n2 ≡ n3 ≡ n mod 2, where n = n1 + n2 + n3 .

2

Proof. Since the Boole coloring of a vertex is the sum of the Boole colorings of its incident edges, we get X v∈V

Ψ(v) =

3 X

ni 1i + (p − n)0 =

i=1

X

2Ψ(e) = 0,

e∈E

but this equality only can hold if either ni 1i = 0 or ni 1i = 1i for alli. Since n1 + n2 + n3 = n, the lemma follows. 2 Corollary 1.2. Let G be a graph with p vertices. Then, there exists no 3-edge-coloring of G with p − 1 vertices Boole colored 0 and the remaining vertex Boole colored 1. 2

2

Edge-Coloring Degree

Given a 3-edge-coloring of G, Ψ, a vertex v ∈ V is said to be conflicting when some of the edges incident to it have the same color. Notice that, when δ(v) = 2, vertex v is conflicting iff Ψ(v) = 0. However, when δ(v) = 3, v is conflicting iff Ψ(v) = 1(1i ). In this case we will say that v is of type 1i . The concept of conflicting vertex allow us to define a parameter that measures how far is G from being of class one: the edge-coloring degree of G, denoted d(G), is the minimum possible number of conflicting vertices that a 3-edge-coloring of G can have. Such a coloring is then called minimal. Of course, G is of class one (respec. class two) iff d(G) = 0 (respec. d(G) ≥ 1). As an example it is easily seen that, for the Petersen graph P , d(P ) = 2. From Corollary 1.2 we have that, for a cubic graph G, d(G) 6= 1. In other words, a cubic graph G is of class two iff d(G) ≥ 2. A conflicting vertex is said to be normalized if only two of its incident edges have the same color. A 3-edge-coloring will be called normalized if its conflicting vertices are all normalized. The following result, proved in [3], shows that it is always possible to get a minimal 3-edge-coloring with all its conflicting vertices normalized. Theorem 2.1. Let G be a graph with edge-coloring degree d(G) = d. Then, there exists a normalized minimal 3-edge-coloring, i.e. a 3-edge coloring with exactly d normalized conflicting vertices. Proof. Clearly, we can assume that G is cubic. Since d(G) = d, there exists a 3-edge-coloring of G, Ψ, with exactly d conflicting vertices, 3

some of which are possibly not normalized. Then it suffices to prove that from Ψ we can obtain another 3-edge-coloring, Ψ′ , with fewer non-normalized vertices. Indeed, assume that (the edges incident to) a vertex v has colors 1,1,1. Then, consider a 1-2 Kempe chain which has v as the initial vertex and goes through the vertices with colors α,β,3 and α,β,β (here a henceforth α,β denote the colors 1,2 in any order, and the underlined colors correspond to the edges of the chain). Since G is finite, this chain necessarily ends in a vertex (perhaps belonging to the chain itself) with colors α,α,α, α,3,3 or α,α,β. Then, by interchanging the colors of the chain, we can obtain in all the cases a new 3-edge-coloring Ψ′ with a number of non-normalized conflicting vertices reduced by one or two. Note that the case α,α,3 is impossible because then Ψ′ would only have d − 1 conflicting vertices against the hypothesis about the value of d(G). This completes the proof. 2 Assume that a graph G, with edge-coloring degree d, contains a 2-factor (i.e. an spanning subgraph made up by disjoint cycles) with r odd cycles. Then there is an obvious 3-edge-coloring of G which has r conflicting vertices (just assign alternatively colors 1,2 to the edges of the cycles and color 3 to the remaining edges). Therefore, d ≤ r, and it is natural to ask in what cases we can have an equality. For instance, that is trivially true for cubic graphs with d = 0 (class one). Since, for cubic graphs, d 6= 1, the next case is that of the following theorem. Theorem 2.2. Let G be any cubic graph with edge-coloring degree d(G) = 2. Then G contains a 2-factor with r = 2 odd cycles. Proof. By Theorem 2.1, there exists a normalized minimal 3-edgecoloring, Ψ, of G. Besides, from Lemma 1.1, the two conflicting vertices of Ψ must be of the same type, say 11 . This means that the colors of the three edges incident to each of these vertices are 1,2,2 or 1,3,3. Then, the required 2-factor is just the subgraph induced by the edges colored 2 and 3. 2 It is natural to think that the larger the edge-coloring degree the larger the order. The following simple result, proved in [2], shows that this is the case for the ”nontrivial” cubic graphs of class two, called snarks. More precisely, a class two cubic graph is called snark if it is cyclically 4-edge-connected and has girth at least 5. See, for instance, Gardner [6] or Isaacs [8]. 4

Theorem 2.3. Let G be a snark with p vertices and edge-coloringdegree d. Then, p ≥ 10 ⌊(d + 1)/2⌋ . Proof. By the well-known result of Petersen [12], G has a 2-factor with a number, say r, of odd cycles. Then, as G has no triangles, p ≥ 5r. Moreover, since G has even order, r is also even. Hence, r ≥ d if d is even and r ≥ d + 1 if d is odd, i.e. r ≥ 2 ⌊(d + 1)/2⌋ and the result follows. 2

3

Critical Graphs

In the study of graphs of class two, a special attention has been paid to the critical graphs, see Fiorini and Wilson [5]. A graph of class two is called critical if the removal of any of its edges lowers its chromatic index. This definition can be generalized in the obvious way by using the concept of edge-coloring degree. Namely, we say that a graph G is c-critical if d(G) = c(≥ 1) and d(G − e) < c for any edge e ∈ E. For instance, P is 2-critical and P − v is (1-)critical. In fact, it is readily seen that, if G is c-critical then d(G − e) = c − 1 for any e ∈ E. Notice that, if G is a graph with edge-coloring degree d(G) = d ≥ 1, then it contains a set of ci -critical subgraphs, i = 1, 2, . . . , n, such that Pn i=1 ci = d. To obtain them it suffices to remove successively all those edges whose deletion does not lower the edge-coloring degree. The next theorem tell us something about the structure of c-critical graphs. The same results were already known for (1-)critical graphs, see [5]. So, cases (ii) and (iii) correspond to the so-called Vizing’s adjacency lemma [13]. Theorem 3.1. Let G be ac-critical graph and let v ∈ V (G). Then, (i) G has no bridges; (ii) if δ(v) = 2 then v is adjacent to two vertices of degree 3; (iii) if δ(v) = 3 then v is adjacent to at most one vertex of degree 2. 2 We omit the proofs since they use the same techniques as in the case of critical graphs. Namely, assume that G has one of the ”forbidden structures”; delete a convenient edge; and show that a minimal 3-edgecoloring of the resulting graph, with c − 1 conflicting vertices, could 5

be then extended to a 3-edge-coloring of G with the same number of conflicting vertices; a contradiction. As a consequence of Theorem 3.1, some properties of critical graphs are shared by the larger family of c-critical graphs. For instance, (i) means that G has no cut-vertices and, therefore, p1 = 0; Moreover, from (ii) and (iii), one can deduce that p2 ≤ p3 /2 and, hence, 4p/3 ≤ q, see [5]. Other conditions on the structure of c-critical graphs are given in the following result: Theorem 3.2. Let G be a c-critical graph. Then, (i) if c = 1 then p2 (G) 6= 0, 2; (ii) if c = 2 then p2 (G) 6= 1. Proof. We will only prove (ii). Case (i) can be proved similarly. So, assume that there exists a 2-critical graph with exactly one vertex, say v, of degree 2 (as said above, p1 = 0). Let u be any other vertex of G, δ(u) = 3. Since G is 2-critical, there exists a 3-edge coloring of G in which the only conflicting vertices are v and u. But, according to Section 1, this implies Ψ(v) = 0, Ψ(u) = 1, and Ψ(w) = 0 for any other vertex w 6= v, w, contradicting Corollary 1.2. 2 The impossibility of the cases c = 1, p2 = 2 and c = 1, p2 = 0 was already known. The former was first proved by Jakobsen in [9], while the latter is a particular case of a result which states that there are no ρ-regular critical graphs with ρ ≥ 3, see [5]. In [9] Jakobsen showed that the so-called Haj´ os-union of two graphs is very useful to obtain (1-)critical graphs of large order (for a list of all critical graphs with order ≤ 10, see [10] or [5]). We will next show that the same construction can be used to obtain c-critical graphs for any given value of c ≥ 1. Let G1 be a c1 -critical graph with a vertex v of degree 2, adjacent to the vertices v1 and v2 ; and let G2 be a c2 -critical graph of which we choose an edge e = w1 w2 . The Haj´os-union of G1 and G2 is then obtained by joining the graphs G1 −v and G2 −e by the new edges v1 w1 and v2 w2 . Hence, the obtained graph has p3 (G1 ) + p3 (G2 ) vertices of degree 3 and p2 (G1 ) + p2 (G2 ) − 1 vertices of degree 2. Moreover we have the following result [3]:

6

Theorem 3.3. In the above conditions, the Haj´ os union, H, of the graphs G1 and G2 is (c1 + c2 − 1)-critical. Proof. Let us see first that d(H) ≥ c1 + c2 − 1. To this end, let us consider the subgraphs H1 = G1 − v and H2 = G2 − e of H. Note that, since Gi is ci -critical, d(Hi ) = ci − 1, i = 1, 2. Let ΨH be a minimal 3-edge-coloring of H. Then ΨH induces a 3-edge-coloring in Hi with at least ci − 1 conflicting vertices. So we can assume that there are exactly ci − 1 conflicting vertices in each Hi . But, according to the edge-coloring degree of G1 (respec. G2 ), the edges v1 w1 and v2 w2 should have equal (respec. different) color, a contradiction. The existence of an edge-coloring ΨH with c1 + c2 − 1 conflicting vertices is a simple consequence of the comments above. To prove that H is (c1 + c2 − 1)-critical, we have to check that d(H − e) < d(H) for any edge e of H. This is clearly true when e is one of the new edges vi wi . Otherwise, i.e. if e belongs to some Hi , the critical quality of H is again a simple consequence of the critical quality of Gi . The details are left to the reader. 2 As an example, the Haj´os union of P (2-critical, p2 = 0) and P − v (1-critical, p2 = 3) produces a 2-critical graph with p2 = 2 vertices of degree two. In our context, the following question is of some interest: Given a pair (c, p2 ) of integers, c ≥ 1, p2 ≥ 0, is there any c-critical graph with (exactly) p2 vertices of degree two?. According to Theorem 3.2, the answer is ”not” for the cases (1,0) (there are no 1-critical cubic graphs), (1,2) and (2,1). The above- mentioned construction allow us to give examples of graphs for all the other cases except for the pair (3,0) which is still unsettled. Problem 3.4. Is there any 3-critical cubic graph? The converse result of Theorem 3.3 also holds [3]: Theorem 3.5. If a c-critical graph G has an edge-cut of two (independent) edges, then there exist two graphs G1 and G2 such that G can be obtained as the Haj´os-union of G1 and G2 . This result gives some orientation to a possible hunter of the graphs described in Problem 3.4: A 3-critical cubic graph, if any, has no edgecuts of two (independent) edges. In other words, any 3-critical graph has maximum edge-connectivity. 7

References [1] Beineke,L.W. & Wilson,R.J., On the edge-chromatic number of a graph, Discrete Math. 5 (1973) 15-20. [2] Fiol,M.A., Contribuci´ on a la Teor´ıa de Grafos Regulares, (Master Dissertation) Universitat Polit`ecnica de Barcelona, 1979. [3] Fiol,M.A., 3′ -grafos cr´ıticos, Research Report, Dpt. de Matem`atica, Universitat Polit`ecnica de Barcelona, 1979. [4] Fiol,M.A., A Boolean algebra approach to the construction of snarks, in Graph Theory, Combinatorics and Applications, vol. 1, (Ed. by Y.Alavi G.Chartrand, O.R.Oellermann and A.J.Schwenk) JohnWiley & Sons, New York (1991) 493-524. [5] Fiorini,S. & Wilson,R.J., Edge-Colouring of Graphs. Research Notes in Mathematics 16, Pitman Publi., London, 1977. [6] Gardner,M., Mathematical Games: Snarks, Boojums and other conjectures related to the four-color-map theorem, Sci. Amer. 234 (1976) 126-130. [7] Harary,F., Graph Theory, Addison-Wesley, Reading, Mass., 1969. [8] Isaacs,R., Infinite families of non-trivial trivalent graphs which are not Tait colorable, Amer. Math. Monthly 82 (1975) 221-239. [9] Jakobsen,I.T., Some remarks on the chromatic index of a graph, Arch. Math. (Basel) 24 (1973) 440-448. [10] Jakobsen,I.T., On critical graphs with chromatic index 4, Discrete Math. 9 (1974) 265-276. [11] Johnson,E.L., A proof of 4-coloring the edges of a cubic graph, Amer. Math. Monthly 73 (1966) 52-55. [12] Petersen,J., Die theorie der regul¨aren graphen, Acta Math., Stockholm, 15 (1891) 193-220. [13] Vizing,V.G., The chromatic class of a multigraph, Kibernetika (Kiev) 3 (1965) 29-39.

8

∗

M.A. Fiol Departament de Matem`atica Aplicada i Telem`atica Universitat Polit`ecnica de Catalunya

Abstract Let G be a (simple) graph with maximum degree three and chromatic index four. A 3-edge-coloring of G is a coloring of its edges in which only three colors are used. Then a vertex is conflicting when some edges incident to it have the same color. The minimum possible number of conflicting vertices that a 3edge-coloring of G can have, d(G), is called the edge-coloring degree of G. Here we are mainly interested in the structure of a graph G with given edge-coloring degree and, in particular, when G is c-critical, that is d(G) = c ≥ 1 and d(G − e) < c for any edge e of G.

1

Edge-colorings

Throughout this paper G = (V, E) denotes a connected (simple) graph with maximum degree ∆ = 3, order p = p(G) = | V (G) | and size q = q(G) = | E(G) |. The degree of a vertex v ∈ V will be denoted by δ(v). For i = 1, 2, 3 let pi = pi (G) denote the number of vertices of G with degree i. Then p1 + 2p2 + 3p3 = 2q and, therefore, p1 ≡ p3 mod 2. A k-edge-coloring (in wide sense) of G, denoted ΨG or Ψ, is simply an assignment of one of k colors to each of its edges. If the edges incident to each vertex have different color we will say that such a ∗

Research supported in part by the Comisi´ on Interministerial de Ciencia y Tecnologia, Spain, under grant TIC90-0712.

1

coloring is proper. The chromatic index of G, χ′ (G), is the minimum integer k such that G has a proper k-edge-coloring. As it is well-known, in our case χ′ (G) must be either 3 or 4; see Johnson [11] or Vizing [13]. In the former case G is said to be of class one. Otherwise, G is said to be of class two. The following characterization of the ”coloring” of a set of edges was used in [2] and [4] to construct snarks. Let G be a graph with a 3-edge-coloring Ψ, and consider a subset of m edges F ⊂ E, mi of which have color i, i = 1, 2, 3. Then we say that the set F has Boole coloring 0, denoted by Ψ(F ) = 0, iff m1 ≡ m2 ≡ m3 ≡ m mod 2. On the other hand, we say that F has Boole coloring 1 (or, more specifically, 1a ), denoted by Ψ(F ) = 1(1a ), iff ma + 1 ≡ mb ≡ mc ≡ m + 1 mod 2 where the letters a, b, c stand for the colors 1, 2, 3 in any order. The Boole coloring of any vertex v ∈ V , denoted by Ψ(v), is then defined to be the Boole coloring of the edges incident to it. It is interesting to note the following facts: when δ(v) = 1, Ψ(v) = 1i iff the edge incident to v has color i, i = 1, 2, 3; when δ(v) = 2, Ψ(v) = 0 (respec. Ψ(v) = 1) iff the two edges incident to v have the same (respec. different) color; when δ(v) = 3, Ψ(v) = 0 iff the three edges incident to v have different colors. It turns out that the set of Boole colorings, {0, 11 , 12 , 13 }, together with the sum operation defined in the natural way, is isomorphic to the Klein group (with 0 as the identity element, 1i + 1i = 0, and 1a + 1b = 1c ), see [4]. Note that, as each element coincides with m)

its inverse, m1i = 1i + · · · +1i is 0 when m is even and 1i when m is odd. From this fact, we have the following useful result which can be seen as a generalization of the so-called parity lemma [8]. Lemma 1.1. Let G = (V, E) be a graph with a given 3-edge-coloring, such that, for i = 1, 2, 3, ni of its vertices have Boole coloring 1i . Then, n1 ≡ n2 ≡ n3 ≡ n mod 2, where n = n1 + n2 + n3 .

2

Proof. Since the Boole coloring of a vertex is the sum of the Boole colorings of its incident edges, we get X v∈V

Ψ(v) =

3 X

ni 1i + (p − n)0 =

i=1

X

2Ψ(e) = 0,

e∈E

but this equality only can hold if either ni 1i = 0 or ni 1i = 1i for alli. Since n1 + n2 + n3 = n, the lemma follows. 2 Corollary 1.2. Let G be a graph with p vertices. Then, there exists no 3-edge-coloring of G with p − 1 vertices Boole colored 0 and the remaining vertex Boole colored 1. 2

2

Edge-Coloring Degree

Given a 3-edge-coloring of G, Ψ, a vertex v ∈ V is said to be conflicting when some of the edges incident to it have the same color. Notice that, when δ(v) = 2, vertex v is conflicting iff Ψ(v) = 0. However, when δ(v) = 3, v is conflicting iff Ψ(v) = 1(1i ). In this case we will say that v is of type 1i . The concept of conflicting vertex allow us to define a parameter that measures how far is G from being of class one: the edge-coloring degree of G, denoted d(G), is the minimum possible number of conflicting vertices that a 3-edge-coloring of G can have. Such a coloring is then called minimal. Of course, G is of class one (respec. class two) iff d(G) = 0 (respec. d(G) ≥ 1). As an example it is easily seen that, for the Petersen graph P , d(P ) = 2. From Corollary 1.2 we have that, for a cubic graph G, d(G) 6= 1. In other words, a cubic graph G is of class two iff d(G) ≥ 2. A conflicting vertex is said to be normalized if only two of its incident edges have the same color. A 3-edge-coloring will be called normalized if its conflicting vertices are all normalized. The following result, proved in [3], shows that it is always possible to get a minimal 3-edge-coloring with all its conflicting vertices normalized. Theorem 2.1. Let G be a graph with edge-coloring degree d(G) = d. Then, there exists a normalized minimal 3-edge-coloring, i.e. a 3-edge coloring with exactly d normalized conflicting vertices. Proof. Clearly, we can assume that G is cubic. Since d(G) = d, there exists a 3-edge-coloring of G, Ψ, with exactly d conflicting vertices, 3

some of which are possibly not normalized. Then it suffices to prove that from Ψ we can obtain another 3-edge-coloring, Ψ′ , with fewer non-normalized vertices. Indeed, assume that (the edges incident to) a vertex v has colors 1,1,1. Then, consider a 1-2 Kempe chain which has v as the initial vertex and goes through the vertices with colors α,β,3 and α,β,β (here a henceforth α,β denote the colors 1,2 in any order, and the underlined colors correspond to the edges of the chain). Since G is finite, this chain necessarily ends in a vertex (perhaps belonging to the chain itself) with colors α,α,α, α,3,3 or α,α,β. Then, by interchanging the colors of the chain, we can obtain in all the cases a new 3-edge-coloring Ψ′ with a number of non-normalized conflicting vertices reduced by one or two. Note that the case α,α,3 is impossible because then Ψ′ would only have d − 1 conflicting vertices against the hypothesis about the value of d(G). This completes the proof. 2 Assume that a graph G, with edge-coloring degree d, contains a 2-factor (i.e. an spanning subgraph made up by disjoint cycles) with r odd cycles. Then there is an obvious 3-edge-coloring of G which has r conflicting vertices (just assign alternatively colors 1,2 to the edges of the cycles and color 3 to the remaining edges). Therefore, d ≤ r, and it is natural to ask in what cases we can have an equality. For instance, that is trivially true for cubic graphs with d = 0 (class one). Since, for cubic graphs, d 6= 1, the next case is that of the following theorem. Theorem 2.2. Let G be any cubic graph with edge-coloring degree d(G) = 2. Then G contains a 2-factor with r = 2 odd cycles. Proof. By Theorem 2.1, there exists a normalized minimal 3-edgecoloring, Ψ, of G. Besides, from Lemma 1.1, the two conflicting vertices of Ψ must be of the same type, say 11 . This means that the colors of the three edges incident to each of these vertices are 1,2,2 or 1,3,3. Then, the required 2-factor is just the subgraph induced by the edges colored 2 and 3. 2 It is natural to think that the larger the edge-coloring degree the larger the order. The following simple result, proved in [2], shows that this is the case for the ”nontrivial” cubic graphs of class two, called snarks. More precisely, a class two cubic graph is called snark if it is cyclically 4-edge-connected and has girth at least 5. See, for instance, Gardner [6] or Isaacs [8]. 4

Theorem 2.3. Let G be a snark with p vertices and edge-coloringdegree d. Then, p ≥ 10 ⌊(d + 1)/2⌋ . Proof. By the well-known result of Petersen [12], G has a 2-factor with a number, say r, of odd cycles. Then, as G has no triangles, p ≥ 5r. Moreover, since G has even order, r is also even. Hence, r ≥ d if d is even and r ≥ d + 1 if d is odd, i.e. r ≥ 2 ⌊(d + 1)/2⌋ and the result follows. 2

3

Critical Graphs

In the study of graphs of class two, a special attention has been paid to the critical graphs, see Fiorini and Wilson [5]. A graph of class two is called critical if the removal of any of its edges lowers its chromatic index. This definition can be generalized in the obvious way by using the concept of edge-coloring degree. Namely, we say that a graph G is c-critical if d(G) = c(≥ 1) and d(G − e) < c for any edge e ∈ E. For instance, P is 2-critical and P − v is (1-)critical. In fact, it is readily seen that, if G is c-critical then d(G − e) = c − 1 for any e ∈ E. Notice that, if G is a graph with edge-coloring degree d(G) = d ≥ 1, then it contains a set of ci -critical subgraphs, i = 1, 2, . . . , n, such that Pn i=1 ci = d. To obtain them it suffices to remove successively all those edges whose deletion does not lower the edge-coloring degree. The next theorem tell us something about the structure of c-critical graphs. The same results were already known for (1-)critical graphs, see [5]. So, cases (ii) and (iii) correspond to the so-called Vizing’s adjacency lemma [13]. Theorem 3.1. Let G be ac-critical graph and let v ∈ V (G). Then, (i) G has no bridges; (ii) if δ(v) = 2 then v is adjacent to two vertices of degree 3; (iii) if δ(v) = 3 then v is adjacent to at most one vertex of degree 2. 2 We omit the proofs since they use the same techniques as in the case of critical graphs. Namely, assume that G has one of the ”forbidden structures”; delete a convenient edge; and show that a minimal 3-edgecoloring of the resulting graph, with c − 1 conflicting vertices, could 5

be then extended to a 3-edge-coloring of G with the same number of conflicting vertices; a contradiction. As a consequence of Theorem 3.1, some properties of critical graphs are shared by the larger family of c-critical graphs. For instance, (i) means that G has no cut-vertices and, therefore, p1 = 0; Moreover, from (ii) and (iii), one can deduce that p2 ≤ p3 /2 and, hence, 4p/3 ≤ q, see [5]. Other conditions on the structure of c-critical graphs are given in the following result: Theorem 3.2. Let G be a c-critical graph. Then, (i) if c = 1 then p2 (G) 6= 0, 2; (ii) if c = 2 then p2 (G) 6= 1. Proof. We will only prove (ii). Case (i) can be proved similarly. So, assume that there exists a 2-critical graph with exactly one vertex, say v, of degree 2 (as said above, p1 = 0). Let u be any other vertex of G, δ(u) = 3. Since G is 2-critical, there exists a 3-edge coloring of G in which the only conflicting vertices are v and u. But, according to Section 1, this implies Ψ(v) = 0, Ψ(u) = 1, and Ψ(w) = 0 for any other vertex w 6= v, w, contradicting Corollary 1.2. 2 The impossibility of the cases c = 1, p2 = 2 and c = 1, p2 = 0 was already known. The former was first proved by Jakobsen in [9], while the latter is a particular case of a result which states that there are no ρ-regular critical graphs with ρ ≥ 3, see [5]. In [9] Jakobsen showed that the so-called Haj´ os-union of two graphs is very useful to obtain (1-)critical graphs of large order (for a list of all critical graphs with order ≤ 10, see [10] or [5]). We will next show that the same construction can be used to obtain c-critical graphs for any given value of c ≥ 1. Let G1 be a c1 -critical graph with a vertex v of degree 2, adjacent to the vertices v1 and v2 ; and let G2 be a c2 -critical graph of which we choose an edge e = w1 w2 . The Haj´os-union of G1 and G2 is then obtained by joining the graphs G1 −v and G2 −e by the new edges v1 w1 and v2 w2 . Hence, the obtained graph has p3 (G1 ) + p3 (G2 ) vertices of degree 3 and p2 (G1 ) + p2 (G2 ) − 1 vertices of degree 2. Moreover we have the following result [3]:

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Theorem 3.3. In the above conditions, the Haj´ os union, H, of the graphs G1 and G2 is (c1 + c2 − 1)-critical. Proof. Let us see first that d(H) ≥ c1 + c2 − 1. To this end, let us consider the subgraphs H1 = G1 − v and H2 = G2 − e of H. Note that, since Gi is ci -critical, d(Hi ) = ci − 1, i = 1, 2. Let ΨH be a minimal 3-edge-coloring of H. Then ΨH induces a 3-edge-coloring in Hi with at least ci − 1 conflicting vertices. So we can assume that there are exactly ci − 1 conflicting vertices in each Hi . But, according to the edge-coloring degree of G1 (respec. G2 ), the edges v1 w1 and v2 w2 should have equal (respec. different) color, a contradiction. The existence of an edge-coloring ΨH with c1 + c2 − 1 conflicting vertices is a simple consequence of the comments above. To prove that H is (c1 + c2 − 1)-critical, we have to check that d(H − e) < d(H) for any edge e of H. This is clearly true when e is one of the new edges vi wi . Otherwise, i.e. if e belongs to some Hi , the critical quality of H is again a simple consequence of the critical quality of Gi . The details are left to the reader. 2 As an example, the Haj´os union of P (2-critical, p2 = 0) and P − v (1-critical, p2 = 3) produces a 2-critical graph with p2 = 2 vertices of degree two. In our context, the following question is of some interest: Given a pair (c, p2 ) of integers, c ≥ 1, p2 ≥ 0, is there any c-critical graph with (exactly) p2 vertices of degree two?. According to Theorem 3.2, the answer is ”not” for the cases (1,0) (there are no 1-critical cubic graphs), (1,2) and (2,1). The above- mentioned construction allow us to give examples of graphs for all the other cases except for the pair (3,0) which is still unsettled. Problem 3.4. Is there any 3-critical cubic graph? The converse result of Theorem 3.3 also holds [3]: Theorem 3.5. If a c-critical graph G has an edge-cut of two (independent) edges, then there exist two graphs G1 and G2 such that G can be obtained as the Haj´os-union of G1 and G2 . This result gives some orientation to a possible hunter of the graphs described in Problem 3.4: A 3-critical cubic graph, if any, has no edgecuts of two (independent) edges. In other words, any 3-critical graph has maximum edge-connectivity. 7

References [1] Beineke,L.W. & Wilson,R.J., On the edge-chromatic number of a graph, Discrete Math. 5 (1973) 15-20. [2] Fiol,M.A., Contribuci´ on a la Teor´ıa de Grafos Regulares, (Master Dissertation) Universitat Polit`ecnica de Barcelona, 1979. [3] Fiol,M.A., 3′ -grafos cr´ıticos, Research Report, Dpt. de Matem`atica, Universitat Polit`ecnica de Barcelona, 1979. [4] Fiol,M.A., A Boolean algebra approach to the construction of snarks, in Graph Theory, Combinatorics and Applications, vol. 1, (Ed. by Y.Alavi G.Chartrand, O.R.Oellermann and A.J.Schwenk) JohnWiley & Sons, New York (1991) 493-524. [5] Fiorini,S. & Wilson,R.J., Edge-Colouring of Graphs. Research Notes in Mathematics 16, Pitman Publi., London, 1977. [6] Gardner,M., Mathematical Games: Snarks, Boojums and other conjectures related to the four-color-map theorem, Sci. Amer. 234 (1976) 126-130. [7] Harary,F., Graph Theory, Addison-Wesley, Reading, Mass., 1969. [8] Isaacs,R., Infinite families of non-trivial trivalent graphs which are not Tait colorable, Amer. Math. Monthly 82 (1975) 221-239. [9] Jakobsen,I.T., Some remarks on the chromatic index of a graph, Arch. Math. (Basel) 24 (1973) 440-448. [10] Jakobsen,I.T., On critical graphs with chromatic index 4, Discrete Math. 9 (1974) 265-276. [11] Johnson,E.L., A proof of 4-coloring the edges of a cubic graph, Amer. Math. Monthly 73 (1966) 52-55. [12] Petersen,J., Die theorie der regul¨aren graphen, Acta Math., Stockholm, 15 (1891) 193-220. [13] Vizing,V.G., The chromatic class of a multigraph, Kibernetika (Kiev) 3 (1965) 29-39.

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