On Graphs with Proper Connection Number 2

An edge-colored graph is properly connected if for every pair of vertices u and v there exists a properly colored uv-path (i.e. a uv-path in which no two consecutive edges have the same color). The proper connection number of a connected graph G, denoted pc(G), is the smallest number of colors needed to color the edges of G such that the resulting colored graph is properly connected. An edge-colored graph is flexibly connected if for every pair of vertices u and v there exist two properly colored paths between them, say P and Q, such that the first edges of P and Q have different colors and the last edges of P and Q have different colors. The flexible connection number of a connected graph G, denoted fpc(G), is the smallest number of colors needed to color the edges of G such that the resulting colored graph is flexibly connected. In this paper, we demonstrate several methods for constructing graphs with pc(G) = 2 and fpc(G) = 2. We describe several families of graphs such that pc(G) ≥ 2 and we settle a conjecture from [3]. We prove that if G is connected and bipartite, then pc(G) = 2 is equivalent to being 2-edge-connected and fpc(G) = 2 is equivalent to the existence of a path through all cut-edges. Finally, it is proved that every connected, k-regular, Class 1 graph has flexible connection number 2.


Introduction
An edge-colored graph is said to be properly colored if no two adjacent edges share a color. An edge-colored graph is said to be properly connected if between every pair of distinct vertices there exists a path that is properly colored. If only two colors are used to color the edges of a graph, we often refer to a properly colored path as an alternating path. The proper connection number of a connected graph G (introduced in [3], [1]), denoted by pc(G), is the minimum number of colors needed to color the edges of G to make it properly connected. Walk and trail analogs of proper connection exist and can be found in [7,13,3].
Properly connected graphs have applications to communication networks, genetics and social sciences. (See for example: [6,5,4].) There are multiple surveys on the topic including one by Bang-Jensen and Gutin [2] that focuses on cycles and paths and includes a section devoted to 2-colored graphs and a dynamic survey on proper connection number maintained by Li and Magnant, [10]. Recently, a text by Li, Magnant, and Qin [11] has been published with paper and electronic versions.
Many previous results have hypotheses concerning the number of bridges; for example, Borozan et al. [3] proved that if the graph G is bridgeless and not a tree, then pc(G) ≤ 3. Another example can be found in Theorem 4.4 [13]. Theorem 4.3 in this manuscript uses a different but equivalent hypothesis concerning bridges to obtain the same conclusion as Theorem 4.4 from [13].
If P is an alternating uv-path in the edge-colored graph G, we use start(P ) and end(P ) to denote the color of the edge incident to u in P and the color of the edge incident to v in P respectively. In several places in [3] and [8], the authors reference an additional "strong property." In this paper, we formally define this property as "flexible." (See Definition 1.1 below.) Note that we do not use the word "strong" since "strong proper path coloring" was defined in [12] to be an edge coloring in which there exists properly colored shortest paths between vertices. Definition 1.1. An edge-colored graph is flexibly connected if between every pair of vertices u and v there exist properly colored paths P and Q such that start(P ) = start(Q) and end(P ) = end(Q).
The flexible connection number of a connected graph G, denoted f pc(G), is the smallest number of colors needed to color the edges of G such that the resulting colored graph is flexibly connected.
Observe that by definition f pc(G) ≥ 2 for all graphs G.
x 0 x 1 x 2 x 3 x 4 x 5 x 6 Figure 1: The 7-cycle above is 2-edge-colored such that it is properly connected. It is not flexibly connected since every path starting at x 0 must begin with red edge. If edge x 0 x 1 was changed to a third color, the 7-cycle would be flexibly connected.
In Section 2 we describe several ways of building properly connected graphs and flexibly connected graphs. In Section 3 we describe several families of graphs that motivate the conjecture below. In Section 4, we characterize bipartite graphs G with pc(G) = 2 and with f pc(G) = 2. In addition, we prove that every connected, k-regular, Class 1 graph has flexible connection number 2 provided k ≥ 2.

Constructions
Here we describe methods of constructing graphs with proper connection number 2 and with flexible proper connection number 2. Construction 2.1 appears as Proposition 2 in [3]. The proof is included for completeness. Construction 2.1. Let G 1 be a 2-edge-colored graph that is properly connected. Let G be obtained from G 1 by adding vertex v and two edges from v to G 1 . Then pc(G) = 2.
Proof. Let G 1 be a 2-edge-colored graph that is properly connected and let v be a vertex added to G 1 such that N (v) = {x, y}. Since the 2-colored graph G 1 is properly connected, there exists an alternating xy-path, P , in G 1 . Color edge xv with the color different from start(P ) and color edge yv with color different from end(P ). Observe that with this coloring, every vertex on P has two alternating paths to v, one along P via x and the other along P via y and these two paths to v begin with different colors.
Since G 1 is properly connected, for every z ∈ V (G 1 ), there exists an alternating zx-path in G 1 , say Q. Observe that Q must share at least one vertex in common with P (perhaps only x). Thus there exists a first vertex shared by both paths Q and P. Now construct an alternating zv-path by following Q to the first vertex it shares with P. Then follow P to v via whichever direction gives a proper coloring.
Construction 2.2. Let G 1 and G 2 be disjoint 2-edge-colored graphs such that G 1 is flexibly connected and G 2 is properly connected. Let G be obtained by adding two independent edges between G 1 and G 2 . Then pc(G) = 2.
Proof. Let u, v ∈ V (G 1 ) and x, y ∈ V (G 2 ) such that ux, vy ∈ E(G). Observe that since G 1 is flexibly connected, it is sufficient to demonstrate a coloring of edges ux and vy such that for every z ∈ V (G 2 ), there exists an alternating zu-path or an alternating zv-path. Such a coloring can be found by adding vertex w to G 2 with edges wx and wy and then applying Construction 2.1 to determine the colors of edges wx and wy. Use these colors on ux and vy to demonstrate that pc(G) = 2.
Construction 2.3. Let G 1 and G 2 be 2-edge-colored graphs such that G 1 is flexibly connected and G 2 is properly connected. Let G be obtained from G 1 and G 2 by identifying a vertex in G 1 with a vertex in G 2 . Then pc(G) = 2.
Proof. Let x be the identified vertex. Since G 2 is properly connected, for every z ∈ V (G 2 ), there exists an alternating zx-path, P , in G 2 . Since G 1 is flexibly connected, for every w ∈ V (G 1 ), there exists an alternating xw-path that begins on the color different from end(P ).
Construction 2.4. Let G 1 be a 2-edge-colored graph that is properly connected and let u, v ∈ V (G 1 ). Let P 1 be an alternating uv-path in G 1 . Let G be obtained from G 1 by adding a uv-path P such that all internal vertices of P are new and such that the cycle obtained from P and P 1 is even. Then pc(G) = 2.
Proof. Alternately color the edges P such that the cycle P ∪ P 1 is alternately colored. Every vertex z in G 1 has a shortest alternating path to some vertex of P 1 . Proceed around the cycle P 1 ∪ P in the direction that makes the coloring proper to find an alternating path to any vertex in P.
Lemma 2.1. If G is a 2-edge-colored graph such that between every pair of vertices u and v there exists a pair of alternating uv-paths, P and Q such that start(P ) = start(Q), then f pc(G) = 2.
Proof. Suppose G is a 2-edge-colored graph such that between every pair of vertices u and v there exists a pair of alternating paths from v to u, say P 1 and P 2 , such that start(P 1 ) = start(P 2 ). If end(P 1 ) = end(P 2 ), then paths satisfying the Lemma have been found. If not, by assumption, there exists an alternating path from u to v, say P 3 , such that start(P 3 ) = end(P 1 ) = end(P 2 ). Since start(P 1 ) = start(P 2 ), the color of end(P 3 ) must be different from one of start(P 1 ) or start(P 2 ) and a pair of paths satisfying the Lemma have been found. Construction 2.5. Let G 1 be a 2-edge-colored graph that is flexibly connected and let u, v ∈ V (G 1 ). Let P 1 be an alternating uv-path in G 1 . Let G be obtained from G 1 by adding a uvpath P such that all internal vertices of P are new and such that the cycle obtained from P and P 1 is even. Then f pc(G) = 2.
Proof. Color the edges of P such that the even cycle obtained from P 1 ∪ P is alternately colored. Lemma 2.1 implies that it is sufficient to show that for every x, y ∈ G there exists a pair of paths from x to y starting on different colors. Let x ∈ G 1 . Since f pc(G 1 ) = 2, we need only consider y ∈ P − {u, v}. Construct the desired paths by selecting two alternating paths from x to P 1 that start on different colors. Each path can be extended to an alternating x, y-path by selecting the direction around the cycle P 1 ∪ P.
Let x be an internal vertex on path P. If y ∈ P 1 ∪ P, a pair of xy-paths starting with different colors can be found by going different directions around the cycle P 1 ∪ P. If y ∈ G 1 − P 1 , then find one xy-path by using xP u and the other using xP v. Since G 1 is flexibly colored, the path to u (respectively v) can be extended to an alternating path to y.

Examples
Observation 3.1. Let u and v be vertices in bipartite graph G. Assume the edges of G are 2-colored. Since the parity of the length of any uv-path in G is fixed, in any alternating uv-path P , the color of start(P ) determines the color of end(P ).
The next example is an infinite family of graphs, each containing roughly half of all possible edges, yet with flexible proper connection number greater than 2.
Example 3.2. Let G be constructed from a complete bipartite graph K m,m by adding one additional vertex u and two edges incident to u with endpoints, v and w, in different partite sets of K m,m . We claim f pc(G) > 2.
Observe that if f pc(G) = 2, then without loss of generality, uv is red and uw is blue. Let x be any other vertex of G. If G is flexibly colored, then there must exist a vx-path, P v , that starts on a blue edge and a wx-path, P w , that starts on a red edge. But the parity of P v and P w are different. Thus, end(P v )=end(P w ). Thus, every alternating xu-path starts on the same color. Thus, f pc(G) > 2.  Example 3.5. The graph G 4 , shown in Figure 4, is 4-regular, 2-edge-connected, and has proper connection number greater than 2. To see that G 4 cannot have proper connection number 2, observe that each triangle induced by vertices {u i , v i , w i } will have at least two monochomatic edges in any 2-coloring of the edges of G 4 . Thus for each i ∈ {0, 1, 2, 3}, at least one of v i or w i will have the property that every path to u i ends with an edge of a fixed color. That is, for one of v i or w i , every alternating path to u i always ends on a red edge or a blue edge. Without loss of generality, assume that for each i, it is the v i u i -path that ends in some fixed color. In order for G 4 to contain an alternating v 0 v 1 -path, the v 0 u 0and v 1 u 1 -subpaths must end in the same color. To have an alternating v 0 v 2 -path, the v 0 u 0and v 2 u 2 -subpaths must end in different colors. But now there can be no properly colored Figure 4: The graph G 4 is a 4-regular, 2-edge-connected graph with proper connection number greater than 2.
From graphs G 4 and G 5 in Examples 3.4 and 3.5 above, we can obtain r-regular, 2-edgeconnected graphs with proper connection number greater than 2 for all r ≥ 6, by appending the appropriate gadget to each vertex. Specifically, to construct a 2k-regular graph with these properties for k ≥ 3, start with G 4 and to each vertex v append the gadget H made from a K 2k+1 from which a matching on k − 2 edges has been deleted. Make v adjacent to the end vertices of the deleted matching. (See Figure 5.) To construct a (2k + 1)-regular graph for k ≥ 3, start with G 5 and to each vertex v append the gadget H made from a K 2k+2 from which a matching on k − 2 edges has been deleted. Make v adjacent to the end vertices of the deleted matching. Example 3.6. For k ≥ 4, there exist graphs G that are 2-edge-connected and k-regular with proper connection number greater than 2.
Next we will demonstrate via an infinite family that there exist 2-connected almost-kregular graphs with proper connection number greater than 2. By "almost-k-regular" we mean every vertex has degree k or k + 1. Note that the graph M is such a graph where k = 2.
In [3] it was conjectured that graphs with connectivity 2 and minimum degree at least 3 have proper connection number 2. This family demonstrates that the conjecture is false.
The construction is easier to understand using the following lemma. Intuitively, this Lemma asserts that if a graph with proper connection number 2 has cut-vertices u 1 and u 2 (see Figure 6) such that the component (shown in the large oval) is bipartite and such that a u 1 u 2 -path contains at least 3 cut-edges in the component (shown in bold), then the whole component can be replaced by a path without changing the proper connection number provided that path has the same parity as one through the component in G. Proof. Assume the edges of G are 2-colored such that G is properly connected. Choose three cut-edges of H on a u 1 u 2 -path in H. Let x 1 x 2 be the middle cut edge of these three edges. Let P and Q be two alternating x i u 1 -paths in H. Since both paths contain at least one cut edge, end(P ) = end(Q). Note that the observation of the previous sentence applies even when paths P and Q start at different x i 's. The same argument applies to any two alternating x i u 2 -paths in H.
Since G is properly connected, for every z ∈ V (G)−V (H), there exists a properly colored x i z-path. Moreover, for each z, the properly colored x i z-path contains a first u j . That is, while an alternating x i z-path may contain both cut vertices, u 1 and u 2 , one must appear first.
With respect to these alternating x i z-paths, there are two possibilities.
Case 1: For at least one x i , every properly colored path to every vertex z ∈ V (G)−V (H) uses the same one of u 1 or u 2 first.
Without loss of generality, assume that for every z ∈ V (G)−V (H), every alternating x 1 zpath exits H via vertex u j for j = 1 or j = 2. It follows that for every z 1 , z 2 ∈ V (G) − V (H), there does not exist an alternating z 1 z 2 -path through C.
Let Q 1 , Q 2 be alternating x 1 z 1 -and x 1 z 2 -paths, respectively. Then each of Q 1 and Q 2 contain alternating x 1 u j -subpaths in H. From our earlier observations end(x 1 Q 1 u j ) = end(x 1 Q 2 u j ), where x 1 Q 1 u j is the subpath of Q 1 starting at x 1 and ending at u j and x 1 Q 2 u j is defined similarly.
We can now demonstrate how to color G such that pc(G ) = 2. Color all edges in G − V (P − {u 1 , u 2 }) the same as in G. Alternately color the edges of P , starting at u j and beginning with color end(x 1 Q 1 u j ).
To demonstrate that pc(G ) = 2, we check pairs of vertices. Since P is alternately colored, any pair of vertices on P are properly connected. Given any two vertices in G − (V (P − {u 1 , u 2 })), the alternating path that existed in G must remain in G since, by assumption in this case, no properly colored path could have used a u 1 u 2 -path through C. Finally, if one vertex, v is in P and one, z, is in G − P , observe that by construction, there exists an alternating path from v to u j . This path arrives at u j using the same color as was used in the alternating x 1 z-path in G. Thus, the path from v to z in G can be completed by continuing along the same u 1 z-path used by x 1 in G.
Case 2: For all i, j there exists some z ∈ V (G) − V (H) such that there exists a properly colored x i z-path through u j .
By assumption, there exists in H an alternating x i u j -path for every i ∈ {1, 2} and every j ∈ {1, 2}. Moreover, for every alternating x 1 u 1 -path Q 1 and every alternating x 2 u 1 -path Q 2 , end(Q 1 ) = end(Q 2 ). The previous statement holds if u 1 is replaced with u 2 . This implies that there exists an alternating u 1 u 2 -path in H, say Q , and start(Q ) = end(Q 1 ) = end(Q 2 ) and end(Q ) must have the same color as every alternating x 1 u 2 -or x 2 u 2 -path in H.
Hence, start(Q ) must be the same color as the edge incident to u 1 in H on every x i z-path for every z ∈ V (G) − V (H).
We can now demonstrate how to 2-color the edges of G such that G is properly connected. Color all edges in G − V (P − {u 1 , u 2 }) the same as in G. Then, alternately color the edges of P starting (and necessarily ending) with the same color as Q .
To demonstrate that pc(G ) = 2, we check pairs of vertices. Since P is alternately colored, any pair of vertices on P are properly connected. Given any two vertices in G − (V (P − {u 1 , u 2 })), if the alternating path in G didn't use vertices from H, the same path exists in G . If the alternating path did use vertices from H, the path must have passed through H. Since every alternating u 1 u 2 -path in H began and ended with the same colors and those colors are used to color P , the path through H in G can be replaced with P in G . Finally, if v ∈ P and z ∈ G − P , recall that an alternating zx 1 -path existed in G. Follow this path from z to u i . By construction, this path must arrive at u i via a color that is compatible with proceeding along P to v.
Observe that graph M is 2-connected, has proper connection number greater than 2, and every vertex has degree 2 or 3. To construct a 2-connected graph G with pc(G) > 2 and such that every vertex has degree 4 or 5, replace each 7-cycle in the graph M with the subgraph in Figure 7 (A) such that w 1 and w 2 in the gadget replace the vertices of degree 3 on each 7-cycle. The result is a graph in which every vertex has degree 4 (white) or degree 5 (black). Two applications of Lemma 3.7 imply that, since M has proper connection number greater than 2, so does G. To construct the general family described in Example 3.8 in Figure 7, each K 4,4 − e can be replaced by K k,k − e. Examples 3.6 and 3.8 indicate that Conjecture 1.1 is sharp. The conjecture certainly holds if G is 3-connected (by Corollary 2 in [3]) or the regularity is sufficiently large (by Lemma 4.4 in [8]).

Main Results
In Theorem 3 of [3], the authors describe an additional "strong property" that we defined as being flexibly connected (Definition 1.1). Restated using this new terminology, Borozan et al. prove that if G is bipartite and 2-connected, then f pc(G) = 2. In the notes after the proof, the authors mention that their proof strategy will still work if the hypothesis "2-connected" is replaced by the hypothesis "2-edge-connected." We give an abbreviated proof of the stronger, 2-edge-connected result from [3] and prove the converse.