Properly Edge-colored Theta Graphs in Edge-colored Complete Graphs

With respect to specific cycle-related problems, edge-colored graphs can be considered as a generalization of directed graphs. We show that properly edge-colored theta graphs play a key role in characterizing the difference between edge-colored complete graphs and multipartite tournaments. We also establish sufficient conditions for an edge-colored complete graph to contain a small and a large properly edge-colored theta graph, respectively.

edge uv ∈ E(G), G − uv is the graph with vertex set V (G) and edge set E(G)\{uv}. For a proper subgraph H of G, we use G − H to denote the graph G − V (H ). An edge-coloring of G is a mapping col : E(G) → N, where N is the set of natural numbers. For a, b ∈ N with a ≤ b, we use [a, b] to denote {i ∈ N | a ≤ i ≤ b}.
A graph G with an assigned edge-coloring is called an edge-colored graph (or throughout this paper simply a colored graph). We say that a colored graph G is a properly colored graph (or PC graph for short) if each pair of adjacent edges (i.e., edges that have precisely one end vertex in common) in G are assigned distinct colors. Let G be a colored graph. For an edge e ∈ E(G), we use col(e) to denote the color of e. For a subgraph H of G, we denote by col(H ) the set of colors that are assigned to the edges of E(H ). The cardinality of col(G) is called the color number of G. We say a color appears (at least k times) at a vertex v ∈ V (G) if it is assigned to at least one (at least k) of the edges incident with v. For a vertex v ∈ V (G), we denote by N c G (v) the set of colors that are assigned to the edges incident with v. We call d c G (v) = |N c G (v)| the color degree of v, and we use δ c (G) = min{d c G (v) | v ∈ V (G)} to denote the minimum color degree of G. When there is no ambiguity, we often write N c (v) for N c G (v) and d c (v) for d c G (v). The existence of PC cycles in different types of colored graphs has been studied extensively during the last decades. Early research on the existence of PC Hamilton cycles in fact dates back to the 1970s [2,4,5,21], but this topic has also attracted new interest more recently [1,19]. Similarly, the existence of PC triangles has been studied by different research groups during the same period as well [6,8,[10][11][12]. These topics have been dealt with for general graphs [6,9,11,12,[16][17][18]23] but also for complete graphs [1,2,[4][5][6][7][8]10,14,15,19,21] and for complete bipartite graphs [1,4,15]. Moreover, the theory involved in the study of PC cycles in edge-colored graph is closely related to the theory of directed cycles in directed graphs. In several proofs of theorems related to PC cycles, the analogy with directed graphs has been applied, and these techniques have often been used in constructions of extremal examples or in dealing with extremal cases. In fact, in this sense edge-colored graphs can be regarded as a generalization of directed graphs. We recall the following constructions for supporting evidence of this view.
u and v belonging to the same partite set, the edge uv is not contained in any PC cycles in G. These observations are implied by the following fact (the proof of which is obvious and omitted).
for all i ∈ [1, ]  In [14], it is revealed that if a colored complete graph G contains no monochromatic edge-cut and there exists a vertex v which is not contained in any PC cycles in G, then a substructure of G is essentially a multipartite tournament. Based on the above observations on the intimate relationship between colored graphs and directed graphs, one may wonder what the actual difference is between these two classes of graphs. This was our main motivation to study the difference between colored complete graphs and multipartite tournaments. It turns out that PC theta graphs play a key role in characterizing this difference. Observation 1.1 clearly implies the following: if a colored complete graph G is essentially a multipartite tournament, then G contains no PC theta graph. On the other hand, it is not difficult to observe that the reverse does not hold in general: not every colored complete graph without PC theta (sub)graphs is essentially a multipartite tournament (see Fig. 1a, b for an example). Before we go into more detail on the difference between colored complete graphs and multipartite tournaments and the role that PC theta graphs play in this, we first introduce another definition and some observations. Definition 1.3 Let G be a colored graph. If δ c (G − S) < δ c (G) for each nonempty proper subset S ⊂ V (G), then we say G is color degree critical (or CD-critical for short).
Obviously, the only CD-critical colored complete graph of minimum color degree 1 is a colored K 2 . We also obtain a clear structure for CD-critical colored complete graphs with minimum color degree 2. This structure is based on the following observation on the existence of small PC cycles in colored complete graphs. Observation 1.2 (Li et al. [14]) Let G be a colored complete graph with δ c (G) ≥ 2. Then G contains a PC cycle of length 3 or 4.

Observation 1.3 Let G be a CD-critical colored complete graph with
Then G is either a PC triangle or one of the graphs in Fig. 1.
Proof Let G be a CD-critical colored complete graph with δ c (G) = 2. Then, by using Observation 1.2, we know that G contains either a PC triangle or a PC cycle of length 4. Since a PC cycle has minimum color degree 2, using Definition 1.3, we conclude that G is either a PC triangle or a colored K 4 containing a PC cycle of length 4. If G ∼ = K 4 and G contains a monochromatic edge-cut, then G must be isomorphic to the graph in Fig. 1a or b. The remaining case is that G ∼ = K 4 , and G contains no monochromatic edge-cut and no PC triangle. It is easy to verify that G must be isomorphic to the graph in Fig. 1c.
Our main result characterizes the difference between CD-critical colored complete graphs and essentially multipartite tournaments in terms of the (non)existence of PC theta graphs, in the following way. In the exceptional case with δ c (G) = 2, G is either isomorphic to the graph in Fig. 1a or to the graph in Fig. 1b Fig. 1a. Given integers t ≥ 3 and n ≥ 2t − 1, let T be a tournament satisfying V (T ) = {v 1 , v 2 , . . . , v n }, δ − (T ) = t − 1 and δ + (T ) ≥ 1. Using Construction 1.1, we can construct a colored complete graph H from T . Let G be the colored complete graph obtained by joining H and F (adding all edges between vertices of H and vertices of F) such that col(uv i ) = i for all u ∈ V (F) and i ∈ [1, n].
In Example 1.1, since H is an induced subgraph of G and δ c (H ) = δ c (G), the graph G is clearly not CD-critical. It is easy to verify that G is not a colored K 4 and that G contains no PC theta graph (by observing that F and H , respectively, contain no PC theta graph, and every edge between F and H is not contained in any PC cycles). It is also easy to check that there is no mapping f : We conclude this introductory section with two other observations that motivated our interest in the existence of PC theta graphs. Let x, y be the two vertices of degree 3 in a PC theta graph Θ k, ,m . Then there are three internally-disjoint PC paths between x and y with starting colors distinct and ending colors distinct. This can be regarded as "local PC connectivity", analogous to the concept of "local connectivity" in undirected graphs (without an edge-coloring). This "local PC connectivity" can help forming larger PC structures, in the following sense. Firstly, consider one PC theta graph H in a colored complete graph G. Assume that P, Q and R are the three internallydisjoint PC paths in H . Then it is easy to verify that for each pair of distinct vertices x, y ∈ V (G) \ V (H ), one of x Py, x Qy and x Ry is a PC path. Secondly, suppose we have vertex-disjoint PC theta graphs H 1 , H 2 , . . . , H k in a colored complete graph G, and let [1, k]. Then, it is again easy to verify that there exists a PC cycle in G containing ∪ k i=1 {x i , y i }. Based on these observations, the existence of PC theta graphs (of small order) might have some implications for finding large PC cycles.
The rest of the paper is organized as follows. In the next section, some additional terminology and notation will be introduced, as well as some auxiliary lemmas that we need for our proof of Theorem 1.4. This proof is presented in Sect. 3.
In Sect. 4, we present and prove the following result, involving a sufficient color number condition for the existence of small PC theta graphs in colored complete graphs. Let G be a colored K n . If |col(G)| ≥ n + 1, then G contains a PC Θ 1,2,2 or a PC Θ 1,2,3 . We also discuss the tightness of the condition.
In Sect. 5, the following color degree condition for the existence of large PC theta graphs is obtained. Let G be a colored K n . If δ c (G) ≥ n+1 2 , then one of the following statements holds: 2 for each vertex u ∈ V (G) and G contains a PC Hamilton cycle; (ii) each maximal PC cycle C in G has a chord uv such that {uv, uC + v, uC − v} is a PC theta graph.
This result is related to the following conjecture of Fujita and Magnant [7]: Let G be a colored K n . If δ c (G) ≥ n+1 2 , then each vertex of G is contained in a PC cycle of length for all ∈ [3, n]. Our result also indicates a possible approach to obtaining results on the existence of PC Hamilton cycles in colored complete graphs. As a consequence, we obtain the following result. Let G be a colored K n . If δ c (G) ≥ n 2 + 1, then each vertex of G is contained in a PC theta graph Θ 1,k,m such that k + m ≥ δ c (G).
We conclude the paper in Sect. 6 with some additional remarks and open questions.

Preliminaries
Let G be a graph and H a subgraph of G. We use G[H ] to denote the subgraph of G induced by V (H ). Let C be a cycle with a fixed orientation. For vertices x, y ∈ V (C), xC + y denotes the segment on C from x to y along the direction specified by the orientation of C, and xC − y the segment on C along the reverse direction. For a vertex v on C, denote by v + and v − the immediate successor and predecessor of v on C, respectively. We set v ++ = (v + ) + and v −− = (v − ) − . Similarly, for vertices x and y on a path P, the segment on P between x and y is denoted by x Py. A graph obtained from two disjoint cycles by joining them by one connecting path, by one edge, or by identifying two vertices is called a generalized bowtie (or g-bowtie for short). See and and we call these sets the dominating set and special dominating set of v in G, respectively. Obviously, for each vertex · · · (a) (b) (c)

Fig. 2 Three different types of g-bowties
We continue by presenting the following lemma that we need as a tool in the later proofs.
Note that we use C 1 and C 2 below (and also C i in the sequel) to denote arbitrary cycles, so C i does not indicate a cycle of length i in this paper. Lemma 2.1 Let H be a PC g-bowtie in a colored complete graph G. Let C 1 , C 2 and P be the two cycles and the connecting path in H with V ( Fig. 3). If G [H ] contains no PC theta graph and there exist an orientation of C 1 and a vertex u on C 1 such that u ∈ Dom G[H ] (u − ), u + ∈ Dom G[H ] (u) and u, u + = x, then for each vertex v ∈ V (C 2 ) \ {y} and each orientation of C 2 , the following three statements hold:

Colored Complete Graphs Without PC Theta Graphs
In this section, we present the proof of Theorem 1.4. For convenience, we repeat the statement of the theorem.

Theorem 1.4 Let G be a CD-critical colored complete graph. Then G contains no PC theta graph if and only if G is essentially a multipartite tournament, unless δ c (G) = 2 and G is a colored K 4 containing a monochromatic edge-cut.
We already noticed that Observation 1.1 implies the following: if a colored complete graph G is essentially a multipartite tournament, then G contains no PC theta graph. So, this establishes the "if" part of Theorem 1.4. For the "only if" part, we deliver the proof in three steps. Lemma 3.1 below deals with the case that δ c (G) ≥ 3 and Based on this structural property, an auxiliary oriented graph is constructed, which helps to find a function f in order to complete the proof of Theorem 1.4. Since the two lemmas require rather long technical proofs, we first present the two lemmas without proofs, and then proceed to use them in order to prove Theorem 1.4. The remaining part of this section is then devoted to the proofs of the two lemmas.
then G is essentially a multipartite tournament.
Next, we show how to prove Theorem 1.4 by applying Lemmas 3.1 and 3.2.
Proof of Theorem 1.4 "⇐ ": As we noticed, the proof is obvious by Observation 1.1.
" ⇒": First, we deal with the case that δ c (G) ≤ 2. As we observed earlier, the only CD-critical colored complete graph with δ c (G) = 1 is a colored K 2 , which is essentially a multipartite tournament. Let G be a CD-critical colored complete graph with δ c (G) = 2. Then by Observation 1.3, G must be a PC triangle or a colored K 4 as in Fig. 1. If G is a PC triangle, then obviously, G is essentially a multipartite tournament. If G ∼ = K 4 and G contains no monochromatic edge-cut, then G is the graph in Fig. 1c , which is essentially a multipartite tournament.
then the proof is completed by directly applying Lemma 3.1. Next, we focus on the case that |col(v, and let the direction of each cycle C i be the direction in D. are not consecutive vertices, then by considering the theta graph {uv, uC + v, uC − v} (See Fig. 4a), we know that col (uv) Fig. 4b). Then C i , C j and the path uv form a PC g-bowtie. By applying Lemma 2.
by considering the following three cases. Fig. 5a), then p ≥ 2. Assume that u p ∈ V (C) for some cycle and col(uv) = f (v). Then u 0 u 1 . . . u i u 0 is a PC cycle. Together with C and the path u i u i+1 . . . u p , we obtain a PC g-bowtie (it is possible that u i = u p = x). Apply Lemma 2.1 to u 1 . We get u 1 / ∈ Dom G (u 0 ), a contradiction.
In this case, assume that Fig. 5b). Apply Lemma 2.1 to v 0 and u 0 , respectively. We get Thus col(v 0 u 1 ) = col(u 0 u 1 ). This contradicts that u 1 ∈ Dom * G (u 0 ). This completes the proof of Theorem 1.4. In the remaining part of this section, we present the proofs of Lemmas 3.1 and 3.2.
Proof of Lemma 3.1 By contradiction. Suppose that G is a counterexample to Lemma 3.1. Since |col(v * , Dom G (v * ))| ≥ 2, we can choose vertices x, y ∈ Dom G (v * ) such that col(v * x) = col(v * y). Let α = col(v * x) and β = col(v * y). Then α and β appear only once at x and y, respectively. Let c 0 = col(x y). Then c 0 / ∈ {α, β}. Considering the symmetry of x and y, without loss of generality, assume that Then {v * , x, y}, S 0 , S 1 , S 2 , . . . , S p form a partition of V (G), where it is not necessary but possible that β ∈ {c i | i ∈ [1, p]} and S 0 = ∅. Since S 0 has a different property than S i (i ∈ [1, p]), we often treat S 0 separately in the sequel. We will prove Lemma 3.1 by analysing the colors between the parts. Our proof is based on a large number of claims, each of which is followed by a proof.
Proof By contradiction. Without loss of generality, suppose that there are vertices v 1 ∈ S 1 and v 2 ∈ S 2 such that col(v 1 v 2 ) = a / ∈ {c 1 , c 2 }. Since {x y, xv * y, xv 2 v 1 y} is not a PC theta graph and the color β appears only once at y, we have col(yv 1 ) = c 1 . By Claim 1, col(yv 1 ) = c 0 . Similarly, we can obtain col(yv 2 ) = c 0 . Consider the theta graph {xv 1 , xv 2 v 1 , xv * yv 1 }. We have a = c 0 . Note that either β = c 1 or β = c 2 . Without loss of generality, assume that β = c 1 . Consider the color of v 1 v * . By Claim 1, we know that col(v 1 However, this implies that {xv * , x yv * , xv 2 v 1 v * } is a PC Θ 1,2,3 , a contradiction. We will obtain a contradiction by proving that δ c (G ) ≥ δ c (G) (this contradicts that G is CD-critical). We do this by first proving two subclaims.

Since T = ∅ and col(y, V (T ))
. By Claims 1 and 2, we know that N c To obtain a final contradiction, it suffices to show that d c . So we can assume that there exists a vertex u j ∈ S j ∩ V (T ) for some j ∈ [1, p] such that col(zu j ) / ∈ {α, c 0 }. By Subclaim 5.1, we have col(yu j ) = c 0 . Recall that col(yv i ) = c i = c 0 . So v i = u j . Considering the theta graph {zu j , zxu j , zv * yu j }, we get col(zu j ) = c j . Since {xv * y, xu j y, xzv i y} is not a PC theta graph and col(zv i ) = c i , we obtain that col(zv i ) = c 0 . However, this implies that {yv * z, yu j z, yv i z} is a PC Θ 2,2,2 , a contradiction.
This completes the proof of Claim 5. Note that Subclaims 5.1 and 5.2, and the assertion that S 0 = Z 0 ∪ Z α ∪ Z β before Subclaim 5.2 are only valid in the proof of Claim 5. Now we continue the proof by three more claims.
We need one more claim before we complete the proof of Lemma 3.1.

Claim 8
The following statements hold: By Claims 2 and 7 Thus col(yu * ) = c i , i.e., col(yu * ) = c 0 (by Claim 1). Apply Claim 3 to the edge yu * . We get Considering Claim 6, we know that the color α appears only once at v * (col(xv * ) = α). Thus, in summary, Claim 8 (i) and (ii) hold, and in particular, . This implies that w ∈ S j for some j ∈ [1, p]. By Claim 8 (i), col(w, V (T )) ⊆ {c 0 , c j }. This forces N c G (w) \ N c G (w) = {c 0 } by the fact that col(xw) = c j . Thus col(yw) = c j and there exists an edge u β v β ∈ T such that col(u β w) = c 0 . Recall that col(yu * ) = c 0 . We obtain that u * = w and that {xv * y, xu * y, xv β u β wy} is a PC Before presenting the proof of Lemma 3.2, we need the following observation. Then V (G) = S ∪ T ∪ {x, y}. Since |col(x, Dom G (x))| = |col(y, Dom G (y))| = 1, we know that col(x, Dom G (x)) = col(y, Dom G (y)) = {α} and the color α appears only once at x and y. Thus Dom G (x) = {y}, Dom G (y) = {x} and for each vertex u ∈ S, both colors col(xu) and col(yu) appear at least twice at u. So we have {x, y, z}). Throughout the proof, the notations x, y, z, α and β always refer to the vertices and colors stated above. Now we deliver the proof by first proving the following claims.  Proof It is equivalent to show that col(zu) ∈ {c 0 , col(yu)} for each vertex u ∈ S, and there cannot exist two vertices u 1 , u 2 ∈ S such that col(zu 1 ) = c 0 and col(zu 2 ) = col(yu 2 ).

Subclaim 1.3
The following statements hold: Proof Consider the colors appearing on the edges incident with z. If β / ∈ col(T − z, {x, y}), then apply Subclaim 1.1 to z and each vertex in T − z. {x, y, z}). We obtain Using Subclaim 1.2, we conclude that either a color number condition for the existence of small PC theta graphs. The proof of the following result is inspired by the proof of Lemma 4.1 in [20].
Proof of Theorem 4.1 By contradiction. Let G ∼ = K n be a counterexample to Theorem 4.1 such that n is as small as possible, and subject to this, |col(G)| is as small as possible. Obviously, n ≥ 5, |col(G)| = n + 1 (otherwise, by merging two colors into a new color, we obtain a counterexample to Theorem 4.1 with a smaller number of colors) and |col(G − S)| ≤ n − |S| for each nonempty subset S of V (G). In particular, when S is a single vertex v, we have |col Then we have Thus |E * | ≥ n −1. Let H be the subgraph of G induced by E * . Then there must exist a vertex x ∈ V (H ) and an integer k ∈ [2, n−1] such that d H (x) = k (since n−1 > n/2). Let u 1 , u 2 , . . . , u k be the neighbors of x in H . Let G 1 = G[{u 1 , u 2 , . . . , u k , x}] and G 2 = G − G 1 . Then G 1 − x is a monochromatic complete graph (otherwise, there exist vertices u, v, w ∈ V (G 1 )\{x} such that col(uv) = col(uw) and {xu, xvu, xwu} is a PC Θ 1,2,2 ). Thus |col(G 1 )| = k + 1 < n + 1 and G 2 is nonempty. We claim that E(G 1 , G 2 ) ∩ E * = ∅. For each edge uv ∈ E(G 1 , G 2 ) with u ∈ V (G 1 ) and v ∈ V (G 2 ), if u = x, then by the construction of G 1 , we know that uv / ∈ E * ; otherwise, u ∈ V (G 1 ) \ {x}. Then, choose a vertex u ∈ V (G 1 ) \ {x, u} (this is doable because k ≥ 2). Since {xu, xu u, xvu} is not a PC theta graph, we have col(uv) ∈ {col(uu ), col(xv)}, and, in particular uv / ∈ E * .
We will complete the proof by distinguishing the following two cases.
The proof of Theorem 4.1 is complete.

PC Theta Graphs of Large Order
In this section, we present a sufficient color degree condition for the existence of large PC theta graphs in colored complete graphs. Our main result of this section is related to the existence of long PC cycles in colored complete graphs, and in particular to the following conjecture due to Fujita and Magnant [7]. [7]) Let G be a colored K n . If δ c (G) ≥ n+1 2 , then each vertex of G is contained in a PC cycle of length for all ∈ [3, n].

Conjecture 5.1 (Fujita and Magnant
In the same paper, they presented a class of colored complete graphs to show that the statement of Conjecture 5.1 would be best possible (the lower bound on δ c (G) cannot be improved), and they proved that each vertex is contained in a PC cycle of length 3, 4, and when n ≥ 13, also in a PC cycle of length 5. Recently, Lo [19] established the existence of a PC Hamilton cycle (a PC cycle of length n) in G for sufficiently large n when δ c (G) ≥ (1/2 + )n for any arbitrarily small constant > 0. In this context, in [15] the following cycle extension theorem was proved.
Theorem 5.2 (Li et al. [15]) Let G be a colored K n and let C be a PC cycle of length k in G. If δ c (G) ≥ max{ n−k 2 , k} + 1, then G contains a PC cycle C * such that V (C) ⊂ V (C * ) and |C * | > k.
Using the result of [7] that each vertex is contained in a small PC cycle, as a corollary of Theorem 5.2, it is easy to obtain the following result. [15]) Let G be a colored K n . If δ c (G) ≥ n+1 2 , then each vertex of G is contained in a PC cycle of length at least δ c (G).

Corollary 5.3 (Li et al.
Interestingly, under the condition δ c (G) ≥ n+1 2 of Conjecture 5.1, even the existence of a PC Hamilton cycle has not been fully verified.
In the remainder of this section, we discuss the existence of PC theta graphs when δ c (G) ≥ n+1 2 . We first need the following natural definition of a maximal PC cycle.
Now we are ready to present the main result of this section.
2 , then one of the following statements holds: 2 for each vertex u ∈ V (G) and G contains a PC Hamilton cycle; (ii) each maximal PC cycle C in G has a chord uv such that {uv, uC + v, uC − v} is a PC theta graph.
Before we give our proof of the above theorem, as a final result of this section we present the following straightforward corollary of Theorem 5.4 and Corollary 5.3 without a proof.

Corollary 5.5
Let G be a colored K n . If δ c (G) ≥ n 2 + 1, then each vertex of G is contained in a PC theta graph Θ 1,k,m such that k + m ≥ δ c (G).
We use the following lemma in our proof of Theorem 5.4.
Proof Suppose, to the contrary, that there does not exist such a PC path. Without loss of generality, assume that i = 1. If col(vv 2 is a PC path satisfying the statement in Lemma 5.6, a contradiction. So we have . Repeating these arguments, switching between u 0 and v, if is even, we obtain col(vv ) = col(v v 1 ) after going along C in one round; if is odd, we obtain col(vv ) = col(v v 1 ) after two rounds. In both cases, we end up with a PC path u 0 Pv 1 v 2 · · · v v satisfying the statement in Lemma 5.6, a contradiction. Now we present our proof of Theorem 5.4.

Proof of Theorem 5.4
Let G be a colored K n satisfying δ c (G) ≥ n+1 2 and suppose that C = v 1 v 2 . . . v v 1 is a maximal PC cycle in G for which statement (ii) of the theorem does not hold. Then, by Theorem 5.2 we know that ≥ δ c (G). Now we generate a set of subgraphs of G using Algorithm 1.
In Algorithm 1, H 0 = C and H i+1 is obtained from H i by adding a vertex u ∈ V (G)\V (H i ) to a vertex v j ∈ V (H i ) such that col(uv j ) / ∈ N c H i (v j ). To limit the possibly many choices for u and v j , we choose a vertex v j with j as small as possible. Since G has a finite number of vertices, Algorithm 1 will eventually stop when no vertex in V (G)\V (H s ) can increase the color degree of any vertex in H s . Algorithm 1 implies the following statements: In this case, T u = ∅. Firstly, we will prove the existence of y 0 . Since the cycle u P u x 0 C − x + 0 u is not a PC cycle, x + 0 = x 1 . Hence we can assume that x 0 C + x 1 = x 0 w 1 w 2 · · · w r x 1 with r ≥ 1 (See Fig. 7). Let α = col(uu 1 ), x 0 = w 0 and x 1 = w r +1 .
If col(uw i ) = α for all i ∈ [1, r ], then by considering the cycle uw r C − x 1 and the fact that col(ux 1 ) = α, we get col(uw r ) = col(w r w r −1 ). Thus α = col(uu 1 ) = col(uw r ) = col(w r w r −1 ). Note that uu 1 , w r w r −1 ∈ E(H ). By the choices of E u and E w r , we know that uw r ∈ E 0 . Thus we can choose y 0 = w r .
The existence proof for y i when i ∈ [1, k] is similar, and therefore omitted.
We continue the proof of Note that for each vertex v ∈ V (H ),