Proper Distinguishing Colorings with Few Colors for Graphs with Girth at Least 5

The distinguishing chromatic number, $\chi_D(G)$, of a graph $G$ is the smallest number of colors in a proper coloring, $\varphi$, of $G$, such that the only automorphism of $G$ that preserves all colors of $\varphi$ is the identity map. Collins and Trenk conjectured that if $G$ is connected with girth at least 5 and $G\ne C_6$, then $\chi_D(G)\le \Delta+1$. We prove this conjecture.

The goal of this note is to prove Theorem 1, which was conjectured 1 by Collins and Trenk [6], and proved for bipartite graphs by Alikhani and Soltani [1]. Theorem 1. If G is connected with girth at least 5 and G = C 6 , then χ D (G) ∆ + 1.
Theorem 1 is best possible in two ways. First, the star K 1,∆ requires ∆ + 1 colors. In fact, Collins and Trenk characterized the trees T for which χ D (T ) = ∆(T ) + 1, and there are an infinite number of these for each ∆. Thus, we cannot improve the upper bound on χ D , even if we require larger girth, or impose some requirement on ∆. Second, for each ∆ 3, they constructed infinitely many graphs G with girth 4, maximum degree ∆, and χ D (G) = 2∆ − 2. Thus, we cannot relax the girth bound.
Key Definition. Let G be a connected graph of girth at least 5, and w ∈ V (G). Let T be a breadth-first spanning tree, rooted at w. Let σ denote the vertex order of T , and for each v ∈ V (G), let σ v denote the set of vertices that precede v in σ. When σ ′ is a prefix of σ, we often slightly abuse notation by writing σ ′ to mean the set of vertices in that order. Let ϕ ′ be a proper coloring of the subgraph induced by some σ ′ . We say color greedily color greedily , with respect to σ and ϕ ′ , to mean that we extend ϕ ′ to a proper coloring ϕ by coloring each v ∈ σ \ σ ′ in order, subject to the following two constraints. (i) If v has a neighbor in σ v (other than its parent), then ϕ(v) is the smallest color not used on If v has no neighbor in σ v (other than its parent), then ϕ(v) is the smallest color not used on σ v ∩ S v , where S v consists of the siblings and parent of v.
We begin with the following easy lemma. Its proof draws on ideas from a similar result in [6], where G has no girth constraint.
Lemma 2. Let G be a connected graph with girth at least 5, and w ∈ V (G). Let T be a breadth-first spanning tree, rooted at w, with vertex order σ. Let ϕ be a greedy coloring of G, with respect to σ and a coloring ϕ ′ of some non-empty prefix σ ′ of σ. If ϕ fixes each vertex in σ ′ , then ϕ is a proper distinguishing coloring.
Proof. By construction, ϕ is proper, so we need only prove that ϕ fixes all vertices. We use induction on |V (G)|, at each step considering the next vertex, v, in σ that is not known to be fixed, along with all of its siblings. By assumption ϕ fixes every vertex in σ ′ . Consider the first vertex, v, that is not yet known to be fixed. By the induction hypothesis, the parent, x, of v is fixed. So, any automorphism must map N(x) to N(x). Suppose that some sibling y of v has a neighbor, z, that is already colored. Since z is already colored, it comes before v and y in T . So, by the induction hypothesis, z is fixed. Since x and z are fixed, and G has no 4-cycle, y is also fixed. So all siblings of v with a non-parent neighbor already colored are fixed. And by construction each sibling of v without such a neighbor already colored has a color different from v (or is fixed, since it is in σ ′ ). So v and all its siblings are fixed, which completes the proof. Lemma 2 plays a key role in our proof of Theorem 1. To illustrate how we use it, we first prove a slightly weaker bound. Proposition 3. If G is connected with girth at least 5, then χ D (G) ∆ + 2.
Proof. Choose an arbitrary vertex w, and let T be a breadth-first spanning tree from w. Let ϕ ′ (w) = ∆ + 2, and let σ ′ = {w}. Now we color greedily, with respect to ϕ ′ and σ. Note that, except on w, this coloring only uses colors from {1, . . . , ∆ + 1}, since each vertex v must avoid the colors either (i) on its neighbors, at most ∆, or (ii) on its parent and siblings, at most 1 + (∆ − 1). Since w is the only vertex colored ∆ + 2, it is fixed. Now ϕ is a proper distinguishing coloring by Lemma 2. Thus, χ D (G) ∆ + 2.

Main Result
Before proving our main result, we need an observation about the largest color possibly used in cases (i) and (ii) of a greedy coloring.
Observation 4. Fix a connected graph G with girth at least 5. Choose an arbitrary w ∈ V (G) and breadth-first spanning tree T , rooted at w. For any prefix σ ′ of σ, proper coloring ϕ ′ of the subgraph induced by σ ′ , and greedy coloring ϕ, we have the following. Each vertex in V (G) \ N(w) colored by (i) uses a color no larger than ∆ + 1 and uses ∆ + 1 only if all of its neighbors are already colored, and all use distinct colors. Each vertex v ∈ V (G) \ N(w) colored by (ii) uses a color no larger than ∆.
Proof. The first statement is obvious. The second holds since v has at most ∆ − 2 siblings.
Proof of Theorem 1. Assume the theorem is false, and G is a counterexample minimizing |V (G)|.
We assume that |V (G)| 3, since otherwise the theorem is trivial. Since G is connected, this implies that ∆ 2. If G is a path, then we color one end of the path with 1, and the remaining vertices with 2 and 3 alternating. If G is a cycle, then we color its vertices (in cyclic order) as 1, 2, 3, 1, 2, . . ., where the vertices after the first five alternate colors 3 and 2. It is easy to check that these colorings are proper and distinguishing, unless G = C 6 .
Suppose, to the contrary, that G has a vertex, w, with degree at most ∆ − 1. Let T w, T be a breadth-first spanning tree rooted at w. Color w with ∆+ 1, then color σ \ {w} greedily; call the resulting coloring ϕ ϕ . Note that no other vertex with degree at most ∆ − 1 uses color ∆ + 1. Thus, w is fixed by ϕ. Now ϕ is a proper distinguishing coloring of G by Lemma 2. Thus, χ D (G) ∆ + 1, a contradiction. Claim 3. G has no vertices w, x, x 3 , with x and x 3 adjacent, such that dist(w, x) 3 and dist(w, x 3 ) = 3. In particular, G has diameter at most 3.
If G has a pair of vertices w and x with dist(w, x) = 4, then let x 3 be a neighbor of x on a shortest path from x to w. Thus, the second statement follows from the first. Now we prove the first. Assume the contrary. Let be the second child of w. Color T greedily except for the following modifications. Color w and x 2 with ∆ + 1, color x 1 and y 1 with 1, and color x 3 after all of its siblings so that the multiset of colors appearing on N(x 2 ) differs from that appearing on N(w). This final step is possible because x is uncolored at the time we color x 3 , so we have at least two options for x 3 . Call the resulting coloring ϕ ϕ . Clearly, ϕ is proper and uses at most ∆ + 1 colors. We must verify that ϕ fixes all vertices. The proof uses Lemma 2, so we must show that ϕ fixes σ ′ . As in Observation 4, no vertex other than w and (possibly) x 2 uses color ∆ + 1 and also has a repeated color in its neighborhood. By construction, the multisets of colors used on N(x 2 ) and N(w) differ. Thus, w is fixed. The only children of w that use a common color are x 1 and y 1 ; so all of its other children are fixed. Consider a child of y 1 , call it y 2 . If y 2 is colored by (ii), then ϕ(y 2 ) ∆, by Observation 4. If y 2 is colored by (i), then y 2 has at most one colored neighbor (other than y 1 ), since G has girth at least 5. Thus, ϕ(y 2 ) 3 ∆. So x 1 has a neighbor (other than w) colored ∆ + 1, but y 1 has no such neighbor. Hence, x 1 and y 1 are fixed. Let σ ′ be the prefix of σ ending with x 2 and ϕ ′ the restriction of ϕ to the subgraph induced by σ ′ . Now Lemma 2 implies that all vertices are fixed, so ϕ is a proper distinguishing coloring using at most ∆ + 1 colors, a contradiction.  Suppose, to the contrary, that ∆ 4 and G has diameter at least 3. Claim 3 implies that G has diameter at most 3, so assume G has diameter 3. Choose w, z 1 , z 2 , z 3 ∈ V (G) w, z i , T such that wz 1 z 2 z 3 is a path and dist(w, z i ) = i for each i ∈ {1, 2, 3}. Let T be a breadth-first spanning tree rooted at w. Each z i is at level i in T . Further, by Claim 3, every neighbor of z 3 is at level 2 in T . Choose x 1 , x 2 , y 1 , y 2 ∈ V (G) x i , y i such that wx 1 x 2 z 3 and wy 1 y 2 z 3 are paths, and x 1 , x 2 , y 1 , y 2 , z 1 , z 2 are distinct; see Figure 2. After possibly reordering some siblings, we can assume that x 1 , y 1 , and z 1 are (respectively) the first, second, and third children of w. We can also assume that z 2 is the first child of z 1 and that z 3 is the last child of z 2 (even though z 3 would naturally be a child of x 2 ). Further, we assume x 2 and y 2 are the second children, respectively, of x 1 and y 1 . Finally, by reordering children of y 1 if needed, we assume that z 2 is not adjacent to the first child of y 1 . Since ∆ 4 (and G has girth at least 5), y 1 has at least 3 children; one of these is y 2 and at most one is adjacent to z 2 , since G has no 4-cycle. Similarly, we assume that y 2 is not adjacent to the first child of x 1 .
Begin by coloring w with 1 and coloring x 1 and z 1 with ∆ + 1. We greedily color the rest of T except for the following modifications; none of these change the order of the vertices, only our choice of color for a vertex when we reach it in the order. No child of x 1 uses color 1 (so its children use colors 2, . . . , ∆). Vertex z 2 uses 1 and no other child of z 1 uses 1. (If such a child, say z ′ , has no colored neighbor other than z 1 , then this happens naturally as a result of a greedy coloring, since z 2 uses 1. If z ′ 2 does have a colored neighbor other than z 1 , then it has at most two: one child of x 1 and one child of y 1 , since G has no 4-cycles. Since ∆ 4, z ′ 2 has some available color from among {2, 3, 4} that is 1 ∆ + 1 2 ∆ + 1 3 4 not already used on any of its neighbors, so z ′ 2 can avoid 1; recall that z 1 uses ∆ + 1 > 4, so z 1 does not forbid a color in {2, 3, 4}.) Note that x 2 and y 2 both use 3. When we color z 3 (as the last child of z 2 ), we do so to ensure that the multisets of colors used on N(w) and N(z 2 ) differ; this is possible because x 2 and y 2 use the same color, so z 3 has at least two available colors. Further, if possible, we also require that z 3 not use color ∆ + 1; so z 3 uses ∆ + 1 only if all of its neighbors use distinct colors, except for x 2 and y 2 , which both use 3. Call the resulting coloring ϕ ϕ . Now we show that ϕ fixes all vertices. Since ϕ(z 2 ) = 1, vertex z 1 is the unique vertex colored ∆ + 1 with at least two neighbors colored 1, by Observation 4. Thus z 1 is fixed. By design, no child of z 1 other than z 2 uses color 1. By our choice of color for z 3 , the multisets of colors on N(w) and N(z 2 ) differ. Thus, w is fixed and z 2 is fixed. Since we already know z 1 is fixed, each neighbor of w is fixed. Now, as in the proof of Lemma 2, by induction, we show for each vertex in N(w) that its children are fixed. Finally, consider a vertex, u, at level 3 of T . By Claim 3, each neighbor of u is at level 2. Thus, since G has no 4-cycle and u has at least 2 fixed neighbors, u is fixed. So χ D (G) ∆ + 1, a contradiction.
Suppose that ∆ 4. By Claim 4, G has diameter 2. Since G is ∆-regular with diameter 2 and girth 5, we know that |V (G)| = ∆ 2 + 1. Choose an arbitrary vertex w w, G ′ , and let be a coloring of G ′ showing this. To extend ϕ ′ to G, use color ∆(G) + 1 on each vertex in N(w) and use an arbitrary color (other than ∆(G) + 1) on w; we call this coloring ϕ ϕ . Clearly w is fixed by ϕ, since each other vertex has at most one neighbor colored ∆(G) + 1. Further, each vertex of G ′ is fixed by ϕ ′ , so we get that each vertex of G ′ is fixed by ϕ. Finally, each vertex v ∈ N(w) has all of its neighbors fixed by ϕ. Since G has no 4-cycles, v is also fixed by ϕ. Thus, χ D (G) ∆(G) + 1, a contradiction. Claim 6. G is either bipartite or vertex-transitive.
Suppose there exist vertices w, x 1 , y 1 w, x 1 , y 1 with x 1 , y 1 ∈ N(w) and no automorphism maps x 1 to y 1 . Let T T be a breadth-first spanning tree, rooted at w, with x 1 and y 1 as the first children of w in T . Color w with ∆ + 1, color x 1 and y 1 with 1, and color the rest of T greedily. Now w is fixed, since it is the only vertex colored ∆ + 1 with two neighbors colored 1. Vertices x 1 and y 1 are fixed, since no automorphism maps one to the other, and all other neighbors of w are fixed, since they receive distinct colors. All remaining vertices are fixed by Lemma 2. So χ D (G) ∆ + 1, a contradiction.
Instead, assume that for every vertex w and x 1 , y 1 ∈ N(w) w, x 1 , y 1 some automorphism maps x 1 to y 1 . Thus, for each pair of vertices x 1 , y 1 joined by a walk of even length, some automorphism maps x 1 to y 1 (the proof is by induction on the length of the walk). If G is bipartite, we are done; so assume it is not. Let C C be an odd cycle in G. Since G the electronic journal of combinatorics 25(3) (2018), #P3.5 is connected, for every pair of vertices x 1 , y 1 ∈ V (G), there exists an x 1 , y 1 -walk of even length. Consider a walk from x 1 to y 1 that visits C. If the walk has odd length, then we extend it by going around C once. Thus, G is vertex-transitive, as desired.
Claim 7. G is either the Petersen graph or the Heawood graph; see Figure 3.
Recall that G is 3-regular, by Claims 2 and 5. Choose an arbitrary vertex w, and let T w, T be a breadth-first spanning tree rooted at w. Since G has girth at least 5, the number of vertices at level 2 is 3(3 − 1) = 6. By Claim 3, G has diameter at most 3, so any additional vertices of G are at level 3. Further, by Claim 3, no pair of vertices at level 3 are adjacent. So each vertex at level 3 has all three of its neighbors at level 2. Since each of the 6 vertices at level 2 has at most two neighbors at level 3, the number of vertices at level 3 is at most 2 3 (6) = 4. Thus, |V (G)| 1 + 3 + 6 + 4 = 14. If G has girth at least 6, then G must be the Heawood graph, since it is the unique 3-regular graph with girth at least 6 and at most 14 vertices. Otherwise, G has a 5-cycle. So Claim 6 implies that G is vertex-transitive.
To finish, we simply check that the only 3-regular vertex-transitive graphs with girth at least 5 and at most 14 vertices are the Petersen graph and the Heawood graph. A convenient reference to verify this is [9], which catalogues all vertex-transitive graphs on at most 19 vertices.

Claim 8. G does not exist.
Consider the 4-colorings in Figure 3 of the Petersen graph (left) and Heawood graph (right). Clearly, both colorings are proper. In the coloring of the Petersen graph, every two vertices with the same color have distinct multisets of colors on their neighborhoods. Thus, the coloring is distinguishing. Now consider the coloring of the Heawood graph. The vertices colored 4 are fixed, since the multisets on their neighborhoods differ. The same is true for the vertices colored 2, except for two with the multiset {1, 1, 3}. But for exactly one of these vertices the neighbor colored 3 has a neighbor colored 4. Thus, the vertices colored 2 are fixed. Now every other vertex has at least two neighbors fixed. Since G has no 4-cycles, every vertex is fixed. So the coloring is distinguishing. This proves claim 8, which finishes the proof of Theorem 1. We conclude with a few remarks. The distinguishing list chromatic number, , is the smallest k such that given any list assignment L with |L(v)| k for all v ∈ V (G), there exists a proper distinguishing coloring ϕ of G such that ϕ(v) ∈ L(v) for all v ∈ V (G). In general, χ ℓ D (G) can be much larger than χ D (G). For example, form a graph G from K 1000,1000 by adding a pendant edge at each vertex and subdividing these edges so the resulting paths have distinct lengths. The only automorphism of G is the identity, so χ D (G) = χ(G) = 2. However, χ ℓ D (G) is at least the list-chromatic number, which is greater than 2 (and grows arbitrarily large on complete bipartite graphs). In contrast, we note that the bound on χ D in Proposition 3 also holds for χ ℓ D .
Proof. The proof is nearly identical to that of Proposition 3. We choose an arbitrary vertex w, choose α ∈ L(w), color w with α and let L ′ (v) = L(v) − α for every other vertex v. Now find a breadth-first spanning tree, T , rooted at w, and color greedily from L ′ with respect to σ. It is straightforward to check that the proof of Lemma 2 still holds in this more general setting.
Observation 6. All proofs in this paper are constructive, and imply algorithms to find the colorings, and these algorithms can be easily implemented to run in polynomial time.
Proof. We can easily check that a graph is regular, and compute a breadth-first spanning tree. To determine diameter, we use an algorithm for all-pairs shortest path. In general, computing the automorphisms of a graph can be difficult. However, we only need to do this for graphs on at most 14 vertices, so we can finish in constant time, simply considering every possible map from V (G) to V (G), and checking whether it is an automorphism.
Question 9. When G is connected with girth at least 5, is χ ℓ D (G) ∆ + 1? Two extensions of list chromatic number are online list chromatic number, denoted χ OL , and correspondence chromatic number, denoted χ corr ; see [10] and [7] for the definitions. When we require also that our colorings be distinguishing, we get the parameters χ OL D and χ corr D . It is natural to study the maximum values of these parameters on certains classes of graphs. In particular, what are the maximum values on connected graphs with girth at least 5? Does the bound of Proposition 5, or even Question 9, still hold?