Odd facial colorings of acyclic plane graphs

Let G be a connected plane graph with vertex set V and edge set E. For X ∈ {V,E, V ∪ E}, two elements of X are facially adjacent in G if they are incident elements, adjacent vertices, or facially adjacent edges (edges that are consecutive on the boundary walk of a face of G). A coloring of G is facial with respect to X if there is a coloring of elements of X such that facially adjacent elements of X receive different colors. A facial coloring of G is odd if for every face f and every color c, either no element or an odd number of elements incident with f is colored by c. In this paper we investigate odd facial colorings of trees. The main results of this paper are the following: (i) Every tree admits an odd facial vertex-coloring with at most 4 colors; (ii) Only one tree needs 6 colors, the other trees admit an odd facial edge-coloring with at most 5 colors; and (iii) Every tree admits an odd facial total-coloring with at most 5 colors. Moreover, all these bounds are tight.


Introduction and Notations
All graphs considered in this paper are simple connected plane graphs provided that it is not stated otherwise. We use standard graph theory terminology according to [2]. However, the most frequent notions of the paper are defined through it. A plane graph is a particular drawing of a planar graph in the Euclidean plane such that no edges intersect except at their endvertices. Let G be a connected plane graph with vertex set V (G), edge set E(G), and face set F (G). The graph is a facial total-coloring such that for every face f and every color c, either no element or an odd number of elements incident with f is colored by c.
The main results of this paper are the following: (i) Every tree admits an odd facial vertexcoloring with at most 4 colors; (ii) Only one tree needs 6 colors, the other trees admit an odd facial edge-coloring with at most 5 colors; and (iii) Every tree admits an odd facial total-coloring with at most 5 colors. Moreover, all these bounds are tight.

Odd facial colorings of trees
A set is odd if it has an odd number of elements, otherwise it is even. Vertices of degree one are leaves. An edge incident to a leaf is a pendant edge.

Odd facial vertex-coloring of trees
A set of vertices is called independent, if no two of its members are adjacent. The vertex set of every tree T on at least two vertices can be decomposed into two independent sets, called partite sets.
Let χ o (G) denote the minimum number of colors required in an odd facial vertex-coloring of a plane graph G. (ii) If χ o (T ) = 3, then T has an odd number of vertices, hence exactly one partite set is odd. On the other hand, if one partite set of T is odd, then by (i) χ o (T ) ≥ 3. If we color the vertices of T in the odd partite set with color 1, color one vertex from the even partite set with color 2, and color all other vertices with color 3, we obtain an odd facial 3-vertex-coloring of T .
(iii) If χ o (T ) = 4, then T has an even number of vertices. From (i) it follows that both partite sets are even. On the other hand, if both partite sets are even, then χ o (T ) ≤ 4 (since every partite set can be decomposed into two odd sets). By (i) and (ii) we have χ o (T ) = 4. Proof. Let c be a facial 3-edge-coloring of T ′ . We extend the coloring c step by step. In each step we color one uncolored edge of T .

Odd facial edge-coloring of trees
First we choose an uncolored edge uv (i.e. an edge from E(T ) − E(T ′ )) which is incident with a vertex of T ′ . W.l.o.g., assume that v ∈ V (T ′ ). Observe that u is incident only with uncolored edges (otherwise T ′ with uv contains a cycle). This implies that uv has at most two facially adjacent edges in T ′ . Consequently, there is an admissible color for uv.
In the next step T ′ ∪ {uv} plays the role of T ′ .
Let χ ′ o (G) denote the minimum number of colors required in an odd facial edge-coloring of a plane graph G.
Moreover, this bound is tight.
Proof. Let T be the tree depicted in Figure 1. It is easy to see that χ ′ o ( T ) = 6. Lemma 2.1 implies that every tree T has a facial 3-edge-coloring. Every facial 3-edge-coloring can be modified to an odd facial edge-coloring using at most 6 colors. If in a facial 3-edge-coloring a color appears on an even number of edges, then we recolor one of them with a new color. Since we recolor at most three edges, the new coloring uses at most six colors.
Any tree in this paper is embedded in the plane. The particular embedding is very important. The tree depicted in Figure 2 with the embedding on the left has an odd facial 2-edge-coloring, and with the embedding on the right, its facial 2-edge-coloring is not odd.  Proof. Assume that c uses the colors 1, 2, 3 and the color 1 appears an odd number of times in T . If the color 2 (resp. 3) appears on an even number of edges, then we recolor one edge of color 2 (resp. 3) with a new color 4 (resp. 5).
Theorem 2.2. Every tree T distinct from T (depicted in Figure 1) has an odd facial 5-edgecoloring.
We distinguish some cases according to the length of P . Case 1. P has at least five edges. Let T ′ be the subtree of T consisting of the first five edges of P . Color the edges v 1 v 2 , v 4 v 5 with color A; the edge v 3 v 4 with color B; and the edges v 2 v 3 , v 5 v 6 with color C. By Lemma 2.1 this coloring of T ′ can be extended to a facial 3-edge-coloring of T . If all colors A, B, C appear an even number of times in T , then we recolor the edges v 1 v 2 , v 3 v 4 , v 5 v 6 with a new color E and we obtain an odd facial 4-edge-coloring. Otherwise we apply Lemma 2.3. Case 2. P has exactly four edges, i.e.
Let Since T ̸ = T , v 2 or v 4 has degree 2. Consequently, T has at most five edges. So it has an odd facial 5-edge-coloring. Case 3. P has exactly three edges, i.e.
In this case, color the edge v 2 v 3 with A and all other edges with B and C so that facially adjacent edges receive different colors. If the color B (resp. C) appears on an even number of edges, then we recolor one edge of color B (resp. C) with a new color D (resp. E). Case 4. The length of P is at most 2.
In this case T is a star. It is easy to see that every star has an odd facial 5-edge-coloring. Note that there are infinitely many trees with χ ′ o (T ) = 3 and also infinitely many trees with χ ′ o (T ) = 5. Let G k be a tree obtained from a path P = v 1 v 2 . . . v 4k+3 on 4k + 3 vertices, k ≥ 0, so that we add 2k + 1 new vertices and join each vertex v 2i , i = 1, 3, . . . , 2k + 1, with one of them, see Figure 3 for illustration. Since the vertices v 2 , v 4 , . . . , v 4k+2 have degree three and they cover all edges of G k , every color appears on 2k + 1 edges in any facial 3-edge-coloring of G k , so χ ′ o (G k ) = 3. Let H k be a tree obtained from G k so that we add two new vertices and join both of them with v 4k+3 , see Figure 3 for illustration. It is not hard to see that H k has no odd facial 3-edge-coloring. Since H k has an odd number of edges, Corollary 2.2 implies that χ ′ o (H k ) = 5.

Odd facial total-coloring of trees
Let χ ′′ o (G) denote the minimum number of colors required in an odd facial total-coloring of a plane graph G. The only tree on three vertices is a path on three vertices. Clearly, it has an odd facial 5-totalcoloring. There are two trees on four vertices. They are depicted in Figure 4 and they also have an odd facial 5-total-coloring. So we can assume that T has at least five vertices. Let P = v 1 v 2 . . . v n−1 v n be a longest path in T . There are two possibilities: either the vertices v 2 and v n−1 have degree two or at least one of them has degree at least three in T . Let T ′ = T − {v 1 , v n } be the tree obtained from T by removing the vertices v 1 and v n . The tree T ′ admits an odd facial total-coloring with five colors, since it has fewer vertices than T . This coloring can be extended to an odd facial 5-total-coloring of T in the following way: First we color the edges v 1 v 2 , v n−1 v n with the same color distinct from the colors of v 2 , v 2 v 3 , v n−2 v n−1 , v n−1 . Thereafter we color the vertices v 1 , v n with the same color distinct from the colors of v 2 , v n−1 , v 1 v 2 . Case 2. v 2 or v n−1 has degree at least three.
Every tree on at least three vertices admits a facial total-coloring with exactly four colors, see [7]. Let c be such a coloring of T with colors 1, 2, 3, 4. In the following we show that c can be modified to an odd facial 5-total-coloring.
First observe that T has an odd number of elements (vertices and edges). Therefore in c one or three colors are used an odd number of times. If three colors are used an odd number of times in c, say 1, 2, 3, then it is sufficient to recolor one element of color 4 with (a new) color 5.
Now assume that only one color, say 1, is used an odd number of times. Case 2.1 T has a pendant edge e = xy of color 1, where x is a leaf.
Since c is a facial total-coloring we have c(x) ̸ = 1, c(y) ̸ = 1, and c(x) ̸ = c(y). Without loss of generality, we can assume that c(x) = 2 and c(y) = 3. In this case it suffices to recolor x with 4 and recolor y with (a new) color 5. Let e 1 = u 1 u 2 and e 2 = u 2 u 3 be two facially adjacent pendant edges in T (such two edges exist because v 2 or v n−1 has degree at least three).

Remarks
Note that odd facial edge-coloring of a plane graph G is closely related to edge decomposition of the dual graph G * into odd subgraphs (subgraphs with all vertices having odd degree), see e.g. [4]. Edge decompositions of graphs into odd (even) subgraphs or characterization of odd (even) factors of graphs have recently drawn a substantial amount of attention, see e.g. [10].