On the total chromatic edge stability number and the total chromatic subdivision number of graphs

A proper total coloring of a graph G is an assignment of colors to the vertices and edges of G (together called the elements of G) such that neighbored elements—two adjacent vertices or two adjacent edges or a vertex and an incident edge—are colored differently. The total chromatic number χ′′(G) of G is defined as the minimum number of colors in a proper total coloring of G. In this paper, we study the stability of the total chromatic number of a graph with respect to two operations, namely removing edges and subdividing edges, which leads to the following two invariants. (i) The total chromatic edge stability number or χ′′-edge stability number esχ′′(G) is the minimum number of edges of G whose removal results in a graphH ⊆ G with χ′′(H) 6= χ′′(G) or with E(H) = ∅. (ii) The total chromatic subdivision number or χ′′-subdivision number sdχ′′(G) is the minimum number of edges of G whose subdivision results in a graph H ⊆ G with χ′′(H) 6= χ′′(G) or with E(H) = ∅. We prove general lower and upper bounds for esχ′′(G). Moreover, we determine esχ′′(G) and sdχ′′(G) for some classes of graphs.


Introduction
We consider finite simple graphs G = (V (G), E(G)). A (graph) invariant ρ(G) is a function ρ : I → R + 0 ∪ {∞} where I is the class of finite simple graphs and R + 0 the set of non-negative real numbers. Invariants mostly are integer-valued. An empty graph is a graph with empty edge set.
It is an interesting topic to determine the stability of an arbitrary invariant ρ(G) of a graph G with respect to two operations, namely removing edges and subdividing edges. The ρ-edge stability number and the ρ-subdivision number of G are defined as follows.
The ρ-edge stability number es ρ (G) of G is the minimum number of edges of G whose removal results in a graph H ⊆ G with ρ(H) = ρ(G) or with E(H) = ∅ (which implies es ρ (G) = |E(G)|).
The ρ-subdivision number sd ρ (G) of G is the minimum number of edges of G to be subdivided such that the resulting graph H fulfills ρ(H) = ρ(G). If such an edge set does not exist, then set sd ρ (G) = |E(G)|.
Note that es χ (G) = sd χ (G) = 0 by definition if G is empty, so we can assume in the following that G is non-empty. Subdividing an edge e = uv of a graph G creates a new graph G e , in which a new vertex w is added and the edge e is removed and replaced by two new edges uw and wv. We write G E for the graph obtained by subdividing all edges of E ⊆ E(G). Note that each edge of E is subdivided exactly once.
These two parameters based on edge removal or edge subdivision, respectively, were discussed in several papers, for example in [2,10] and more recently in [1,3,[7][8][9] for the invariants chromatic number χ(G) and chromatic index χ (G).
In this paper we investigate the χ -edge stability number es χ (G), also called total chromatic edge stability number, and the χ -subdivision number sd χ (G), also called total chromatic subdivision number, with respect to the total chromatic number χ (G) of G. Results on the total chromatic number are collected for example in [11] and first results on sd χ (G) can be found in [8].
A (proper) total coloring of G is an assignment of colors to the vertices and edges of G (together called the elements of G) such that neighbored elements-two adjacent vertices or two adjacent edges or a vertex and an incident edge-are colored differently. A k-total coloring is a proper total coloring with k colors. The total chromatic number χ (G) of G is defined as the minimum k in a k-total coloring of G.
We prove in this paper among others general upper and lower bounds for the total chromatic edge stability number es χ (G) (general bounds for the total chromatic subdivision number sd χ (G) can be found in [8]). We also exactly determine es χ (G) for specific classes of graphs such as acyclic graphs, cycles, complete graphs, and complete bipartite graphs. Moreover, we determine sd χ (G) for some graph classes, for example for even complete graphs and for complete bipartite graphs, extending results of [8].

General results for the total chromatic edge stability number
Some general results for the ρ-edge stability number are proved in [7]. If we apply the results on the total chromatic number, then we obtain the following statements. General proofs can be found in [7].
Theorem 12 of [7] states that the edge stability number for specific invariants can be computed by the edge stability numbers of its components. Applying this theorem to the total chromatic edge stability number gives the following result. Proposition 2.2. Let G = H 1 ∪ · · · ∪ H k be the disjoint union of the subgraphs H 1 , . . . , H k whose indices and the integer s, Therefore, we can assume in the following without loss of generality that G is connected. The following results from [7] give lower bounds based on not necessarily distinct subgraphs of a graph. Theorem 2.1. Let G be a non-empty graph with χ (G) = k. If G contains s non-empty subgraphs G 1 , . . . , G s with χ (G 1 ) = · · · = χ (G s ) = k such that a ≥ 0 is the number of edges that occur in at least two of these subgraphs and q ≥ 1 is the maximum number of these subgraphs with a common edge, then both es Corollary 2.2. Let G be a non-empty graph with χ (G) = k. If G contains s non-empty subgraphs G 1 , . . . , G s with χ (G 1 ) = · · · = χ (G s ) = k and pairwise disjoint edge sets, then es Note that es χ (H) ≤ es χ (G) does not hold in general. The ρ-edge stability number es ρ (G) for ρ(G) = ∆(G) where ∆(G) is the maximum degree of G was determined in [7].
is the subgraph of G induced by V ∆ , and α (G) is the edge independence number or matching number of G.
Proof. If G is non-empty, then there is a set E of edges of G such that |E | = es χ (G) and It follows that ∆(G − E ) < ∆(G) which implies that es χ (G) = |E | ≥ es ∆ (G).
Let α (G) be the total independence number of G, that is, the maximum number of pairwise not adjacent or incident elements in G. Let t (G) be the minimum number of elements in a color class of G where the minimum is taken over all total colorings of G with χ (G) colors. Let s be the minimum number of vertices of maximum degree in a color class with t (G) elements if G is of type 1 and s = 0 if G is of type 2.
Proof. Let G be non-empty and consider a total coloring of G with χ (G) colors such that C is a color class with t (G) elements and moreover s vertices of maximum degree if G is of type 1.
If C does not contain any vertex, then removing all edges of C reduces the total chromatic number, thus es χ (G) ≤ t (G).
If C contains vertices, then these will be recolored in an arbitrary order. Let v ∈ C. If there is a missing color in the closed neighborhood N (v) ∪ {v} of v, say color β, then removing the at most one edge of color β incident to v allows us to recolor vertex v by color β. This is for example the case if χ (G) > ∆(G) + 1 or if d(v) < ∆(G). If otherwise there is no missing color in the closed neighborhood of v, then this implies that χ (G) = ∆(G) + 1, d(v) = ∆(G), and the neighbors of v are colored pairwise differently. Select a color β different from the color of v and remove the edge of color β incident to v as well as the edge connecting v with the neighbor of color β. After removing these two edges, vertex v can be recolored by color β. Repeat this recoloring for all other vertices of C (which are independent). Removing the edges of C again reduces the total chromatic number, which implies that es χ (G) ≤ t (G) + s.
By the pigeonhole principle, any total coloring of G with χ (G) colors has a color class with at most |V (G)|+|E(G)| On the other hand, the lower bound implies the last inequality of the statement of the theorem.
In the next section it will be shown that for complete graphs K n of odd order, which are of type 1, es χ (K 3 ) = 3 > α (K 3 ) = 2 holds, that is, the upper bound α (G) + s is tight for K 3 , and es χ (K n ) = α (K n ) = (n + 1)/2 if n ≥ 5 is odd. The question arises for which type 1 graphs G the upper bound for es χ (G) can be improved to α (G).

Theorem 2.2. If G is a type 2 graph and the Total Coloring Conjecture is true, then es
Proof. Since G is of type 2, the graph G is not empty and the invariants ∆(G), type(G) = 2, and χ (G) = ∆(G) + 2 can be reduced by edge removal. By which implies |E | ≥ es χ (G). By removing es type (G) edges E such that type(G − E ) = 1 we obtain Yap ([11], p. 6) stated that if χ (G) = t and if for any t-total coloring of G every color class contains at least two edges of G, then χ (G − e) = t for every edge e of G, that is, es χ (G) ≥ 2. This implies that a graph G with es χ (G) = 1 must have a t-total coloring with a color class with at most one edge.

Total chromatic edge stability number for specific graph classes
In this section we determine the total chromatic edge stability number of some well-known graph classes. Proposition 3.1. For a path P n it holds that es χ (P n ) = n − 1.
Proof. If n ≥ 2 then χ (P n ) = 3 and moreover χ (P n − E ) = 3 for any set of edges E E(P n ), since P n − E contains a subgraph P 2 with total chromatic number 3. Therefore, es χ (P n ) = |E(P n )| = n − 1.

Proposition 3.2.
For a cycle C n it holds that es χ (C n ) = n if 3 | n and es χ (C n ) = 1 if 3 n.
On the other hand, if 3 n, then C n − e ∼ = P n with χ (C n − e) = 3 < 4 = χ (C n ) for any edge e of the cycle, which implies that es χ (C n ) = 1.
By Propositions 2.2 and 3.1, es χ (G) = |E(G)| if ∆(G) ≤ 2, so assume in the following that ∆(G) ≥ 3. Since G is a type 1 graph, es χ (G) ≥ es ∆ (G) by Lemma 2.1. On the other hand, let E be a set of edges of G such that |E | = es ∆ (G) and by the minimality of the total chromatic edge stability number.
This result implies that the lower bound of Lemma 2.1 is tight if G is an acyclic graph with ∆(G) ≥ 3. On the other hand, the difference between es χ (G) and es ∆ (G) may be arbitrarily large, e.g., for cycles of orders divisible by 3 or for unions of paths of order at least 3. For example, if G ∼ = k P 3 , then es χ (G) = |E(G)| = 2k and es ∆ (G) = k.
We now consider complete graphs K n .
Theorem 6.24 of [11] states that if G is a graph of odd order n = 2k [5] proved the following result on the total chromatic number of complete graphs of even order from which edges are removed:

Hilton
Let n ≥ 2 even, E ⊆ E(K n ), j be the maximum number of independent edges in E . Then χ (K n − E ) = n + 1 if and only if |E | + j ≤ n/2 − 1.
The last result of this proposition shows that also for type 2 graphs G the difference between es χ (G) and es ∆ (G) may be arbitrarily large, since es ∆ (K n ) = n/2 for n ≥ 2.
A wheel W n , n ≥ 3, is the join of a cycle C n and a singleton K 1 . The graph W n − v 1 z is also of type 1: At first color the elements of W n − v 1 z as above, then recolor edge v 1 v n and vertex z by color n and vertex v n by color n − 2, in order to obtain a proper n-total coloring of W n − v 1 z. It follows that es χ (W n ) = 1 if n ≥ 4.
For complete bipartite graphs K a,b it holds that K a,b is of type 1 if a = b and of type 2 if a = b. In order to determine the total chromatic edge stability number of K a,b we use the following lemma. Proof. Color the edges of the bipartite graph with colors 1, . . . , ∆(G) which is possible by the Theorem of König (see [4], p. 257). Color the vertices of the partition set containing all vertices of maximum degree by color ∆(G) + 1. Every vertex v of the second partition set has a degree smaller than ∆(G), thus there is a color from {1, . . . , ∆(G)} missing at its incident edges. This color can be used to color v since the vertices of a partition set are pairwise non-adjacent.
Note that this condition is not necessary, since, for example, C 6 is a regular bipartite graph of type 1. Proposition 3.6. For a complete bipartite graph K a,b , a ≤ b, it holds that es χ (K 1,2 ) = 2, es χ (K a,b ) = a if a < b,  (a, b) = (1, 2), and es χ (K a,a ) = a/2 .
2. If a < b, then K a,b is of type 1 by Lemma 3.1, thus es χ (K a,b ) ≥ es ∆ (K a,b ) = a by Lemma 2.1.
If a ≥ 3 is odd then we need to interchange two edge colors in c. Note that the edges incident with u 2 are colored by 2, . . . , a + 1, that is, color 1 is missing at u 2 , and that the edge v (a+1)/2 u 2 is colored by color (a + 3)/2. At vertex u (a+5)/2 , color (a + 3)/2 is missing and the edge v (a+1)/2 u (a+5)/2 is colored by color 1, that is, we can interchange the colors of these two edges incident to vertex v (a+1)/2 . After this recoloring, edge v (a+1)/2 u 2 is colored by 1, edges v i u i+2 are colored by color 2i + 1 for i ∈ {1, . . . , (a − 1)/2}, and edges v i u i+1 are colored by 2i − a − 1 for i ∈ {(a + 3)/2, . . . , a}, that is, these a edges are again colored by pairwise different colors {1, . . . , a}, and the proof can be analogously completed as in the case of even a.
Let E be a set of edges such that |E | = es χ (K a,a ) and χ (K a,a − E ) < χ (K a,a ) = a + 2. Denote by V 1 , V 2 the partition sets of K a,a and let X 1 ⊆ V 1 and X 2 ⊆ V 2 be the sets of vertices incident to the edges of E . Without loss of generality, assume |X 1 | ≤ |X 2 |.
Consider a total coloring of K a,a − E with a + 1 colors. Each color class contains at most a elements (say for each v ∈ V i either v or an edge incident to v) and possibly some additional vertices of X 3−i if the color class does not contain vertices of V i \ X i , i ∈ {1, 2}. Taking all color classes into account, there may be at most |X 1 | vertices added in such a way, therefore the number of elements in all a + 1 color classes is at most (a + 1)a + |X 1 |. On the other hand, K a,a − E has 2a vertices and a 2 − |E | edges, so it must hold that (a + 1)a + |X 1 | ≥ 2a + a 2 − |E | which is equivalent to |X 1 | + |E | ≥ a. Since |X 1 | ≤ |E |, we obtain the lower bound es χ (K a,a ) = |E | ≥ a/2.
We consider the following edge coloring of K a,a : c(v i u j ) = 1 + (l + j − i) mod a, i, j ∈ {1, . . . , a}.
The edges {v l+j u j : j = 1, . . . , l} if a is even or {v l+j u j : j = 1, . . . , l − 1} if a is odd, respectively, are colored by color 1. We recolor these edges by color a + 1 and color the vertices of V 1 \ X 1 = {v l+1 , . . . , v a } by 1 and the vertices of . . , u a } by a + 1.
In the following we determine a set E of l independent and pairwise distinctly colored edges from the subgraph K l,l of K a,a induced by X 1 ∪ X 2 .
After removing the l selected edges of E , their respective end-vertices can be colored by the color of the removed edge. If a is odd, then vertex u l will be recolored by color a + 1. This completes a total coloring of K a,a − E with only a + 1 colors. It follows that es χ (K a,a ) ≤ |E | = a/2 .
Observe that the above proof is constructive. The described total coloring of K a,a − E will be used in Proposition 4.6. Let us note that it is also possible to prove the case es χ (K a,a ) = a/2 using a theorem of Hilton [6] on complete bipartite graphs. This result states that if E ⊆ E(K a,a ) and j is the maximum number of independent edges in E , then χ (K a,a − E ) = a + 2 if and only if |E | + j ≤ a − 1.
A generalized θ-graph θ l1,...,lm , l 1 ≤ · · · ≤ l m , is a graph with two vertices connected by m internally disjoint paths of length l 1 , . . . , l m . If m = 1, then θ l1 is a path of length l 1 and if m = 2, then θ l1,l2 is a cycle of length l 1 + l 2 which have been discussed in the above propositions, so in the following we may assume m ≥ 3. Note that the two vertices of maximum degree m are adjacent if and only if l 1 = 1. Proof. In [8] it was shown that all generalized θ-graphs with m ≥ 3 paths are of type 1, thus es χ (G) ≥ es ∆ (G) by Lemma 2.1, where es ∆ (G) = 1 if l 1 = 1 and es ∆ (G) = 2 if l 1 > 1.
1. l 1 = 1: Denote by e the edge connecting the two vertices of maximum degree. If m ≥ 4, then removing e gives a generalized θ-graph with m − 1 ≥ 3 paths which is of type 1. If m = 3 and l 2 + l 3 is divisible by 3, then removing e gives a cycle C l2+l3 which is also of type 1. Therefore, χ (G − e) = ∆(G − e) + 1 = m < m + 1 = χ (G), that is, es χ (G) = 1 in these two cases.
If m = 3 and l 2 + l 3 is not divisible by 3, then removing a single edge gives a cycle C l2+l3 of type 2 or a graph of maximum degree 3, that is, the total chromatic number does not decrease, and es χ (G) ≥ 2 follows. On the other hand, removing e and a second arbitrary edge gives a path P l2+l3 which is of type 1, thus es χ (G) = 2 holds in this case.
2. l 1 > 1: If m ≥ 4 then removing the first and the last edge of one of the m paths of G gives the union H of a path and of a generalized θ-graph with m − 1 ≥ 3 paths which is of type 1. If m = 3 then removing two independent edges incident with a vertex of maximum degree each gives a path H which is of type 1. In either case, χ (H) = ∆(H) + 1 = m < m + 1 = χ (G) which implies that es χ (G) ≤ 2 and therefore es χ (G) = 2.

Total chromatic subdivision number
The total chromatic subdivison number sd χ (G) of a graph G is the minimum number of edges of G whose subdivision results in a graph H with χ (H) = χ (G). If such an edge set does not exist, then sd χ (G) = |E(G)| (see Introduction). This invariant was first studied in [8]. In that paper sd χ (G) was determined for several classes of graphs but the determination for type 2 graphs remained nearly completely open. For example, it was proved that sd χ (G) = |E(G)| if G is acyclic. For cycles C n it holds that sd χ (C n ) = 1 if n ≡ 0 (mod 3) or n ≡ 2 (mod 3), and sd χ (C n ) = 2 if n ≡ 1 (mod 3). If G is a generalized θ-graph consisting of m ≥ 3 paths, then sd χ (G) = |E(G)|. For complete graphs K n it holds that sd χ (K 1 ) = 0 and sd χ (K n ) = 1 if n = 2 or n ≥ 3 is odd. Note that K n is of type 1 if n is odd.
We will apply the following proposition for studying sd χ (G) for type 2 graphs. The result says that if the Total Coloring Conjecture holds for a graph G, then it also holds for the resulting graph after subdividing an arbitrary number of edges of G. We give a direct proof of this known result.
Proof. If ∆(G) ≤ 2 then G consists of components which are paths or cycles. Subdividing edges of G gives longer paths or cycles. Since the Total Coloring Conjecture holds for paths and for cylces, it holds for G and for G E .
Therefore, we may assume in the following that ∆(G) ≥ 3, which implies that ∆(G E ) = ∆(G). Let e = uv be an arbitrary edge of G and w be the subdivision vertex in G e . Consider a (∆(G) + 2)-total coloring of G. Color all elements of G e except w, uw, vw as in G, then color uw and vw by different colors missing at u and at w, respectively, which is possible since at most ∆(G) colors were used at u, w. Finally, color w by a color different from the colors of the 4 incident or adjacent elements which is possible since there are ∆(G) + 2 ≥ 5 available colors. Therefore, χ (G e ) ≤ ∆(G) + 2.
Proof. If χ (G) does not change by edge subdivisions, then es χ (G) ≤ |E(G)| = sd χ (G). This holds for ∆(G) = 1, therefore we may assume in the following that ∆(G) ≥ 2 which implies that ∆(G E ) = ∆(G) for any set E of edges.

Proof.
1. By definition, sd χ (K 2 ) = 1. Consider as an example the complete graph K = K 4 , say with vertices v 1 , v 2 , v 3 , v 4 , and the graph K E obtained by subdividing the edge of , v 4 by color 3, and v 1 v 4 , v 2 v 3 , w 1 by color 4 gives a total coloring of K E with ∆(K E ) + 1 = 4 colors. It follows that sd χ (K 4 ) = 1. This coloring will be generalized in the proof below.
Our aim is to select a set E of (k + 1)/2 independent edges which are colored pairwise differently with even colors. With the given edge coloring we can select the edges E = {v 2t−1 v 2t : 1 ≤ t ≤ (k + 1)/2 } which are obviously independent. Their colors are C = {c(v 2t−1 v 2t ) : 1 ≤ t ≤ (k + 1)/2 } = {4t − 2 : 1 ≤ t ≤ (k + 1)/2 } which are all colors congruent to 2 modulo 4 from 2 to 2k − 2 if k is even or from 2 to 2k if k is odd, respectively. Note also that each edge from E connects consecutive vertices, thus their respective missing colors are consecutive odd numbers or 2k − 1 and 1 for the last edge v k v k+1 if k is odd. In any case, the two missing colors are different.
We now consider the graph K E in which edge v 2t−1 v 2t of E is subdivided by vertex w t , 1 ≤ t ≤ (k + 1)/2 . We extend the edge coloring c to a total coloring c of K E as follows: Color each non-subdivided edge as in c. Color v 2t−1 w t and w t v 2t with the missing color of v 2t−1 and of v 2t , respectively, which are different, for 1 ≤ t ≤ (k + 1)/2 . Color the vertices v 2t−1 and v 2t with the even color c(v 2t−1 v 2t ) = 4t − 2 and color w t with a possible of the 2k colors which exists for k ≥ 2 since the neighbored elements of w t only use 3 colors, 1 ≤ t ≤ (k + 1)/2 . Color the remaining vertices v k+1+(k mod 2) , . . . , v 2k with the corresponding missing colors which are odd and pairwise different by construction.
Proof. If a < b, then K a,b is of type 1 with no adjacent vertices of maximum degree. Thus Proposition 4.4 implies that sd χ (K a,b ) = |E(K a,b )| with the possible exception of K 2,3 . Since K 2,3 ∼ = θ 2,2,2 , it also holds for this graph that sd χ (K 2,3 ) = |E(K 2,3 )| by Proposition 22 of [8] where the total chromatic subdivision number of generalized θ-graphs is determined.
Consider now K a,a which is of type 2, thus, by Propositions 3.6 and 4.2, es χ (K a,a ) = a/2 ≤ sd χ (K a,a ). It holds that sd χ (K 1,1 ) = sd χ (K 2 ) = 1 and sd χ (K 2,2 ) = sd χ (C 4 ) = 2 (see above). For a ≥ 3 we will use the notations and the described (a + 1)-total coloring of K a,a − E of the proof of Proposition 3.6, where |E | = l = a/2 . This is a partial total coloring of (K a,a ) E where only the subdivided edges and the subdivision vertices are still uncolored.
For each edge e = v i u j ∈ E with subdivision vertex w i different colors are missing at the end-vertices (namely a + 1 at v i and 1 at u j except if j = l and a ≥ 5 is odd, in which case (3l + 2 − l mod 2)/2 is missing) which can be used to color the edges v i w i and w i u j , respectively. Note also that w i is neighbored to four elements, that is, if a ≥ 4 then there is an available color among the a + 1 ≥ 5 colors of the total coloring of K a,a − E which can be used to color w i . If a = 3, then w i can be colored by color 3. This completes the (a + 1)-total coloring of (K a,a ) E . Therefore, χ ((K a,a ) E ) = a + 1 and sd χ (K a,a ) ≤ |E | = a/2 follows.

Concluding remarks
The parameters es ρ (G) and sd ρ (G) were studied in [7,8] especially for the invariants chromatic number χ(G) and chromatic index χ (G). It was shown that es χ (G) = sd χ (G) for all graphs except for bipartite graphs G with cycles, for which es χ (G) = |E(G)| and sd χ (G) = 1 hold. This difference is caused by the fact that the chromatic number of even cycles increases after one edge subdivision. Moreover, es χ (G) = sd χ (G) for all graphs G with χ (G) = ∆(G) + 1. This means that in these cases it does not matter whether edge removals or edge subdivisions will be carried out.