Improper interval edge colorings of graphs

A $k$-improper edge coloring of a graph $G$ is a mapping $\alpha:E(G)\longrightarrow \mathbb{N}$ such that at most $k$ edges of $G$ with a common endpoint have the same color. An improper edge coloring of a graph $G$ is called an improper interval edge coloring if the colors of the edges incident to each vertex of $G$ form an integral interval. In this paper we introduce and investigate a new notion, the interval coloring impropriety (or just impropriety) of a graph $G$ defined as the smallest $k$ such that $G$ has a $k$-improper interval edge coloring; we denote the smallest such $k$ by $\mu_{\mathrm{int}}(G)$. We prove upper bounds on $\mu_{\mathrm{int}}(G)$ for general graphs $G$ and for particular families such as bipartite, complete multipartite and outerplanar graphs; we also determine $\mu_{\mathrm{int}}(G)$ exactly for $G$ belonging to some particular classes of graphs. Furthermore, we provide several families of graphs with large impropriety; in particular, we prove that for each positive integer $k$, there exists a graph $G$ with $\mu_{\mathrm{int}}(G) =k$. Finally, for graphs with at least two vertices we prove a new upper bound on the number of colors used in an improper interval edge coloring.


Introduction
A proper t-edge coloring of a graph G is called an interval t-coloring if the colors of the edges incident to every vertex v of G form an interval of integers. This notion was introduced by Asratian and Kamalian [3] (available in English as [4]), motivated by the problem of constructing timetables without "gaps" for teachers and classes. Generally, it is an NP-complete problem to determine whether a bipartite graph has an interval coloring [25]. However some classes of graphs have been proved to admit interval colorings; it is known, for example, that trees, regular and complete bipartite graphs [3,15,20], bipartite graphs with maximum degree at most three [15], doubly convex bipartite graphs [2,21], grids [11], and outerplanar bipartite graphs [12] have interval colorings. Additionally, all (2, b)-biregular graphs [15,16,22] and (3,6)-biregular graphs [6] admit interval colorings, where an (a, b)-biregular graph is a bipartite graph where the vertices in one part all have degree a and the vertices in the other part all have degree b.
Improper (or defective) colorings was first considered independently by Andrews and Jacobson [1], Harary and Jones [17], and Cowen et al. [7]. This coloring model is a well-known generalization of ordinary graph coloring with applications in various scheduling and assignment problems, see e.g. the recent survey [26], or [7].
Motivated by scheduling and assignment problems with compactness requirements, but where a certain degree of conflict is acceptable, we consider improper interval edge colorings in this paper. An improper edge coloring of a graph is called an improper interval (edge) coloring if the colors on the edges incident with every vertex of the graph form a set of consecutive integers. This edge coloring model seems to have been first considered by Hudak et al. [18], although their investigation has a different focus than ours.
Note that unlike the case for interval colorings, every graph trivially has an improper interval edge coloring. An improper interval coloring is k-improper if at most k edges with a common endpoint have the same color. We denote by µ int (G) the smallest k such that G has a k-improper interval edge coloring. The parameter µ int (G) is called the interval coloring impropriety (or just impropriety) of G.
Improper interval edge colorings have immediate applications in scheduling problems, where an optimal schedule without waiting periods or idle times is desirable, but a certain level of conflict is allowed. For a bipartite graph G, representing a scheduling problem, the parameter µ int (G) has a natural interpretation as the minimum degree of conflict necessary in a schedule with no waiting periods. Moreover, in view of the fact that not every graph has an interval coloring, the parameter µ int (G) may be viewed as a natural measure of how far from being interval colorable a graph is.
Trivially, if G has an interval coloring, then µ int (G) = 1. In this paper, we provide several families of graphs with large impropriety; in particular, we prove that for each positive integer k, there is a graph G with µ int (G) = k.
We prove general upper bounds on µ int (G) and determine µ int (G) exactly for some families of graphs G; in particular we prove that for any graph G with ∆(G) ≥ 6, where ∆(G) and δ(G) denotes the maximum and minimum degree of a graph G, respectively; if G is bipartite and has no vertices of degree three, and for any bipartite graph G; • µ int (G) ≤ r 2 if G is a complete r-partite graph. Furthermore, we conjecture that outerplanar graphs have impropriety at most 2 and we prove this conjecture for graphs with maximum degree at most 8. Finally, we consider the number of colors in an improper interval edge coloring and obtain a new upper bound on the number of colors used in such a coloring.

Preliminaries
The degree of a vertex v of a graph G is denoted by d G (v). ∆(G) and δ(G) denote the maximum and minimum degrees of G, respectively. For two positive integers a and b with a ≤ b, we denote by [a, b] the interval of integers {a, . . . , b}.
We shall need a classic result from factor theory. A 2-factor of a multigraph G (where loops are allowed) is a 2-regular spanning subgraph of G.
Theorem 2.1. (Petersen's Theorem). Let G be a 2r-regular multigraph (where loops are allowed). Then G has a decomposition into edge-disjoint 2-factors.
If α is an edge coloring of G and v ∈ V (G), then S G (v, α) (or S (v, α)) denotes the set of colors appearing on edges incident to v; the smallest and largest colors of the spectrum S (v, α) are denoted by S (v, α) and S (v, α), respectively.
The chromatic index χ ′ (G) of a graph G is the minimum number t for which there exists a proper t-edge coloring of G. A graph G is said to be Class 1 if χ ′ (G) = ∆(G), and Class 2 if χ ′ (G) = ∆(G) + 1. The next result gives a sufficient condition for a graph to be Class 1 (see, for example, [10]). Theorem 2.3. If G is a graph where no two vertices of maximum degree are adjacent, then G is Class 1.
Every bipartite graph is Class 1, as the following well-known proposition, known as König's edge coloring theorem, states.
We shall also need some preliminary results on interval edge coloring. The following was proved by Hansen [15].
Theorem 2.5. If G is a bipartite graph with maximum degree ∆(G) ≤ 3, then G has an interval coloring.

Improper interval edge colorings of some non-intervalcolorable graphs
In this section we determine the impropriety of some well-known families of graphs that in general do not admit interval colorings; in particular we describe constructions of bipartite graphs with arbitrarily large impropriety.

The impropriety of some non-interval-colorable graphs
Regular Class 1 graphs are trivially interval colorable, while no Class 2 graphs are [3,5,14]; however, all regular graphs have small impropriety.
Proof. Let G be a regular graph. It is well-known that G is interval colorable if and only if G is Class 1. Hence, it suffices to prove that µ int (G) ≤ 2; we shall give an explicit 2-improper interval coloring of G. Suppose first that the vertex degrees of G are even, say d G (v) = 2k for every vertex v ∈ V (G). By Petersen's theorem G has a decomposition into 2-factors F 1 , . . . , F k . By coloring all edges of F i by color i, i = 1, . . . , k, we obtain a 2-improper interval coloring of G.
Suppose now that d G (v) = 2k − 1 for all v ∈ V (G). By taking two copies G 1 and G 2 of G and adding an edge between corresponding vertices of G 1 and G 2 , we obtain a 2k-regular supergraph H. By the preceding paragraph, H has a 2-improper interval coloring. By taking the restriction of this coloring to G 1 , it follows that µ int (G) ≤ 2.
Note that Proposition 3.1 implies that for cycles C n (n ≥ 3) and complete graphs K n it holds that µ int (C n ) = µ int (K n ) = 1, if n is even, 2, if n is odd.
Next, we consider generalizations of two families of bipartite graphs with no interval colorings introduced by Giaro et al. [13]. For any a, b, c ∈ N, define the graph S a,b,c as follows: . , x a , y 1 , . . . , y b , z 1 , . . . , z c } and Figure 1 shows the graph S 7,7,7 . Next we define a family of graphs M a,b,c (a, b, c ∈ N). We set Clearly, S a,b,c and M a,b,c are connected bipartite graphs. Giaro et al. [13] showed that the graphs S k = S k,k,k and M l = M l,l,l do not admit interval colorings if k ≥ 7, and l ≥ 5, respectively.
Here we shall prove that all graphs in the families {S a,b,c } and {M a,b,c } satisfy that µ int (S a,b,c ) ≤ 2 and µ int (M a,b,c ) ≤ 2, respectively.  Proof. Without loss of generality, we may assume that a ≤ b ≤ c.
We first construct an edge coloring α of the graph S a,b,c . We define this coloring as follows: (2) for 1 ≤ j ≤ b, let α (u 0 y j ) = α (u 2 y j ) = j; (3) for 1 ≤ k ≤ c, let α (u 0 z k ) = α (u 3 z k ) = a + k; It is straightforward that α is a 2-improper interval coloring of S a,b,c . Next we define an edge coloring β of the graph M a,b,c as follows: (4 ′ ) for 1 ≤ j ≤ b, let β (u 2 y j ) = β (u 3 y j ) = j + 1; (5 ′ ) for 1 ≤ k ≤ c, let β (u 0 z k ) = a + k; It is easy to verify that β is a 2-improper interval coloring of M a,b,c . We conclude that µ int (S a,b,c ) ≤ 2 and µ int (M a,b,c ) ≤ 2.
Lastly, let us consider two elementary classes of graphs that have been proved not to always admit interval colorings. Recall that a wheel graph W n on n vertices (n ≥ 4) is defined as the join of C n−1 and K 1 . It is well-known that only few wheels are interval colorable, but they all have small impropriety (which in fact is implicit in [18]). Proposition 3.3. If W n is a wheel graph on n vertices, then Proof. Let W n be a wheel graph. In [5,14], it was shown that W n has an interval coloring if and only if n = 4, 7 or 10. Hence, it suffices to prove that µ int (W n ) ≤ 2; this follows from a result in [18]: in fact the improper interval (n − 1)-coloring of W n described in the proof of Theorem 2.8 in [18] is a 2-improper interval coloring of W n .
In [8], the authors considered the problem of constructing interval edge colorings of socalled generalized θ-graphs; a generalized θ-graph, denoted by θ m , is a graph consisting of two vertices u and v together with m internally-disjoint (u, v)-paths, where 2 ≤ m < ∞. These graphs also have small impropriety. Proof. In [8], it was proved that θ m has an interval coloring if and only if it is not an Eulerian graph with an odd number of edges. Hence, it suffices to prove that µ int (θ m ) ≤ 2; for i = 1, . . . , m, we color all edges of the ith path between u and v by color i. Thus, trivially µ int (θ m ) ≤ 2.

Graphs with large impropriety
In this section we describe several families of graphs with large impropriety. We begin our considerations with constructions based on subdivisions.
Let G be a graph and V (G) = {v 1 , . . . , v n }. Define graphs S(G) and G as follows: In other words, S(G) is the graph obtained by subdividing every edge of G, and G is the graph obtained from S(G) by connecting every inserted vertex to a new vertex u. Note that S(G) and G are bipartite graphs.
Theorem 3.5. If G is a connected graph and where P is a set of all shortest paths in S(G) connecting vertices w ij , then µ int ( G) > k.
Proof. Suppose, to the contrary, that G has a k-improper interval t-coloring α; Consider the vertex u, and let w and w ′ be two vertices adjacent to u satisfying that α(uw) = S(u, α) = s and α(uw which is a contradiction.
Our next construction uses techniques first described in [24] and generalizes the family of so-called Hertz graphs first described in [13].
Let T be a tree and let P be the set of all paths in T . We set , and define M(T ) as follows: Now let us define the graph T as follows: Clearly, T is a connected graph with ∆( T ) = |F (T )|. Moreover, if T is a tree in which the distance between any two pendant vertices is even, then T is a connected bipartite graph.
Proof. Suppose, to the contrary, that T has a k-improper interval t-coloring α for some t ≥ |F (T )| k . Consider the vertex u. Let v and v ′ be two vertices adjacent to u such that α(uv) = S(u, α) = s and α(uv From this, we have and thus |F (T )| ≤ k (M(T ) + 2), which is a contradiction.
Corollary 3.9. If T is a tree in which the distance between any two pendant vertices is even and |F (T )| > k (M(T ) + 2), then the bipartite graph T has no k-improper interval coloring.
The deficiency of a graph G is the minimum number of edges whose removal from G yields a graph with an interval coloring. Thus, the deficiency of a graph is another measure of how far from being interval colorable a graph is.
As mentioned above, our constructions by trees generalize the so-called Hertz's graphs H p,q , first described in [13]. Hertz's graphs are known to have a high deficiency, so let us specifically consider the impropriety of such graphs.
In [13] the Hertz's graph H p,q (p, q ≥ 2) was defined as follows: The graph H p,q is bipartite with maximum degree ∆(H p,q ) = pq and |V (H p,q )| = pq+p+2. We are now able to prove the following result; our main result of this section. Proof. For a given k, choose p so that p ≥ 2k 2 − 1. Let us consider the tree T = H p,k − d.
Since M(T ) = p + 2k, |F (T )| = pk and the graph H p,k is isomorphic to T , by Theorem 3.8, we obtain that µ int (H p,k ) > k − 1. On the other hand, let us define an edge coloring α of H p,k as follows: It is easy to verify that α is a k-improper interval coloring of H p,k ; thus µ int (H p,k ) ≤ k.
In the last part of this section we use finite projective planes for constructing bipartite graphs with large impropriety. This family of graphs was first described in [24].
Proof. Suppose, to the contrary, that the graph G = Erd(r 1 , . . . , r n 2 +n+1 ) has a k-improper interval t-coloring α for some t ≥ If l i 0 = l j 0 , then, by the construction of G there exists k 0 such that k 0 v If, on the other hand l i 0 = l j 0 , then l i 0 ∩ l j 0 = ∅; so again, by the construction of G, there exists k 0 such that k 0 v and thus Hence, r i ≤ 2k(n + 1), which is a contradiction.

Upper bounds on the impropriety of graphs
In this section, we give general upper bounds on µ int (G) for several different families of graphs. There is a prominent line of research on interval colorings of bipartite graphs; we begin this section by considering improper interval colorings of bipartite graphs.

Bipartite graphs
As mentioned above, Hansen [15] proved that if G is bipartite and satisfies that ∆(G) ≤ 3, then G has an interval coloring, while the question of interval colorability for bipartite graphs of maximum degree 4 is open. However, using Hansen's result and König's edge coloring theorem, we deduce the following upper bound.
Proof. Let G be a bipartite graph. To prove (i), we construct a new bipartite graph H from G by proceeding in the following way: for every vertex v of degree at least δ(G) + 1, we split v into as many vertices of degree δ(G) as possible, and one vertex of degree less than δ(G).
Since the graph H has maximum degree δ(G), by König's edge coloring theorem, it has a proper δ(G)-edge coloring ϕ. Let ϕ G be the coloring of G induced by this coloring of H. Since each vertex of G is split into at most ∆(G) vertices, the coloring ϕ G is a ∆(G) δ(G) -improper interval coloring of G using δ(G) colors.
Part (ii) can be proved similarly to part (i), except that we apply Theorem 2.5 to the graph obtained from G by splitting every vertex of G into vertices of degree at most three.
If G is bipartite, and, in addition, has no vertices of degree 3, then we have the following: Proof. We proceed as in the preceding proof. From the bipartite graph G, we construct a graph G ′ by splitting every vertex of degree at least five into as many vertices of degree four as possible, and one vertex of degree at most three. From G ′ , we construct a graph G ′′ with even vertex degrees by taking two copies of the graph G ′ and joining any two corresponding vertices of degree three or one by an edge. Finally, we construct a 4-regular multigraph H by adding a loop at every vertex of degree two. Now, by Petersen's theorem, H has a decomposition into two 2-factors F 1 and F 2 . In G ′′ , the subgraph F i corresponds to a collection of even cycles, i = 1, 2. By coloring the edges of every cycle in G ′′ corresponding to a cycle of F 1 alternately by colors 1, 2; and the edges of every cycle corresponding to a cycle of F 2 alternately by colors 3, 4, we obtain an interval edge coloring ϕ of G ′′ , where every vertex of degree 2 has colors 1 and 2, or 3 and 4, on its incident edges.
Since there are no vertices of degree three in G, and each vertex of G is split into at most For bipartite graphs with small vertex degrees we deduce some consequences of the above results.  In general, for k ≥ 2, it would be interesting to determine or bound the least integer f bip (k) for which there exists a graph G with maximum degree f bip (k) satisfying µ int (G) = k. Even the case k = 2 of this problem is open. It is known, however, that 4 ≤ f bip (2) ≤ 11, see e.g. [24]. Moreover, by the results of Hertz graphs, f bip (3) ≤ 51, and by the above corollary f bip (3) ≥ 7.

General graphs
Let us now deduce some upper bounds for general graphs. As for bipartite graphs, we define f (k) as the smallest integer such that there exists a graph G with maximum degree f (k) and µ int (G) = k. The smallest graphs with impropriety 2 are odd cycles; thus f (2) = 2.
We believe that the following question is of particular interest: , that is, determine the least integer ∆, such that there is a graph G with maximum degree ∆ satisfying µ int (G) = 3.
The following result shows that f (3) > 5 in Problem 4.5.
Proof. If G has maximum degree 2, then trivially µ int (G) ≤ 2. Let us now consider the case when G satisfies 3 ≤ ∆(G) ≤ 4; again, we shall use Petersen's 2-factor theorem. From G we form a new graph G ′ by taking two copies of G and adding an edge between any two corresponding vertices of odd degree. From G ′ we form a new 4-regular graph H by adding a loop at every vertex of degree 2 in G ′ . By Petersen's theorem, H has a decomposition into two 2-factors F 1 and F 2 . In G ′ , F i corresponds to a collection A i of cycles, i = 1, 2. By coloring edges of all cycles of A i by color i, we obtain a 2-improper interval coloring ϕ of G ′ , and the result now follows by coloring G according to the restriction of ϕ to one of the copies of G in G ′ . Let us now consider the case when ∆(G) = 5. Let G 5 be the subraph of G induced by the vertices of degree 5 in G. Let M be a maximum matching in G 5 . Since M is maximum, the graph H = G − M either has maximum degree 4 or no two vertices of degree 5 in H are adjacent. It follows that H has a proper 5-edge coloring; in the former case by Vizing's theorem, and in the latter case H is Class 1 by Theorem 2.
Since M is a maximum matching in G 5 , the coloring β is a 2-improper interval 3-coloring of G − M ′ . From β we define an edge coloring γ of G as follows: for every uv ∈ E(G), let If there is an edge e 0 such that γ(e 0 ) = 0, then we define an edge coloring γ ′ of G as follows: γ ′ (e) = γ(e) + 1 for every e ∈ E(G). It is straightforward that if this holds, then γ ′ is a 2-improper interval 4-coloring of G; otherwise γ is a 2-improper interval 3-coloring of G. Thus, µ int (G) ≤ 2.
We note that the upper bound in Theorem 4.6 is in fact sharp, since any regular Class 2 graph is not interval colorable.
It also seems that graphs G whose vertex degrees are sufficiently concentrated satisfy µ int (G) ≤ 2; for instance, as pointed out above, any regular graph G satisfies that µ int (G) ≤ 2. We strengthen this observation slightly as follows.
Proof. Let G be a graph satisfying ∆(G) − δ(G) ≤ 1, and let G ∆ be the subraph of G induced by the vertices of maximum degree in G. By the preceding proposition, we may assume that ∆(G) Using the preceding proposition, we can prove the following, by splitting vertices.
Proof. We proceed as before: from G we form a new graph G ′ by splitting every vertex of degree at least δ(G) + 1 into as many vertices of degree exactly δ(G) as possible, and one vertex of degree at most δ(G). Let H be a δ(G)-regular supergraph of G ′ . By Proposition 4.7, H has a 2-improper interval edge coloring. This coloring induces a 2 ∆(G) δ(G) -improper interval coloring of G.
Finally, we have the following general upper bound. Proof. Since any regular graph has impropriety at most 2, it suffices to consider the case when δ(G) < ∆(G). Furthermore, without loss of generality, we assume that G is connected.
Let H be the graph obtained by taking two copies of G and adding an edge between any two corresponding vertices of odd degree. We shall consider G as a subgraph of H.
Since all vertex degrees in H are even, it has an Eulerian circuit T . By coloring all edges of T by 1 and 2 alternately along T , we obtain an improper interval coloring ϕ of H. Let α be the improper interval coloring of G induced by ϕ. If every vertex of G is incident with at most ⌈ ∆(G) 2 ⌉ edges with the same color, then the desired result follows, so assume that this does not hold. Then there is a vertex v which is incident with exactly ⌈ ∆(G) 2 ⌉ + 1 edges with the same color under α, say 1. Indeed, since the edges of T are colored alternately by colors 1 and 2, v must be the first vertex of the Eulerian circuit T in H. Without loss of generality, we assume that v is a vertex of maximum degree in G.
Let us first consider the case when ∆(G) is odd, that is, ∆(G) = ∆(H) − 1. Let T 1 be a shortest subtrail of T from v to a vertex u of degree at most ∆(G) − 1 in G. We define a new coloring ϕ ′ of H from ϕ by recoloring the edges on T 1 in the following way: we set if ϕ(e) = 2 and e ∈ E(T 1 ), 2, if ϕ(e) = 1 and e ∈ E(T 1 ), ϕ(e), if e / ∈ E(T 1 ).
Since d G (u) ≤ ∆(G) − 1, and all vertices of H except v and u are incident with equally many edges of color i under ϕ ′ as under ϕ, i = 1, 2, it follows that the restriction of ϕ ′ to G is a ⌈ ∆(G) 2 ⌉-improper interval coloring of G.
Let us now consider the case when ∆(G) is even, i.e. ∆(H) = ∆(G). Let T 1 be a shortest subtrail of T from v to a vertex x of degree at most ∆(G) − 1 in G, and suppose e 1 is the last edge of T 1 . We consider some different cases. (c) If e 1 ∈ E(G), d G (x) = ∆(G) − 1, ϕ(e 1 ) = 1 (2), and x is incident with an edge e 2 = e 1 in H of color 1 (2) under ϕ that is not in G, then we proceed as follows: let T 2 be the subtrail of T beginning with v whose last edge is e 2 . By proceeding as in (a) and switching colors on T 2 , and taking the restriction of the obtained coloring to G, we obtain a ⌈ ∆(G) 2 ⌉-improper interval coloring of G.

Outerplanar graphs
In this section we consider outerplanar graphs. We do not know of any outerplanar graph G with µ int (G) ≥ 3; in fact, we believe that there is no such graph. Since there are examples of outerplanar graphs with no interval edge coloring, the upper bound in Conjecture 4.10 would be sharp if true. Next, we shall prove that this conjecture holds for graphs with maximum degree at most eight. For the proof of this result, we shall use the well-known fact that an outerplanar graph is Class 1 unless it is an odd cycle [9].
Proof. Since a graph G has a k-improper interval coloring if every block of G has a k-improper interval coloring, it suffices to consider the case when G is 2-connected.
Consequently, assume that G is a 2-connected; then it has a Hamiltonian cycle C. The graph G − E(C) has maximum degree at most 6. If |V (C)| is even, then we define a proper edge coloring ϕ of C by coloring its edges alternately by colors 2 and 3. If |V (C)| is odd, then we define ϕ in the following way: it is well-known that every 2-connected outerplanar has a vertex v of degree 2; we color the edges of C alternately by colors 2 and 3, and beginning and ending with color 2 at the edges incident with v. Now, consider the graph H = G − E(C). Since H is an outerplanar graph (or consisting of several outerplanar components), it has a proper edge coloring α using at most 6 colors 1, . . . , 6. From α, we define a new edge coloring α ′ by recoloring any edges of colors 5 and 6 by colors 1 and 4, respectively. It is straightforward to verify that the colorings ϕ and α ′ taken together form a 2-improper interval coloring of G.
Using the same vertex splitting technique as several times before, we deduce the following corollary. Note that if G is outerplanar and v ∈ V (G), then given integers k and l such that k + l = d G (v), it is always possible to split the vertex v into two new vertices v ′ and v ′′ of degrees k and l, respectively, so that the resulting graph is outerplanar (or a union of vertex-disjoint outerplanar graphs). We state this observation as a lemma.
Lemma 4.12. If G is outerplanar, v is a vertex of G and k and l are positive integers satisfying d G (v) = k + l, then we can split the vertex v into two new vertices of degrees k and l, respectively, in such a way that the resulting graph is outerplanar (or a union of vertex-disjoint outerplanar graphs). Proof. By Proposition 4.11, we may assume that ∆(G) ≥ 9. As in the preceding proof, it suffices to consider the case when G is 2-connected. Let C be a Hamiltonian cycle of G; we color C as in the proof of Proposition 4.11. The result now follows by splitting all vertices of G − E(C) into as many vertices of degree 4 as possible, and possibly one additional vertex of degree at most 3; by repeatedly applying Lemma 4.12, this can be done so that the resulting graph J is outerplanar (or a union of disjoint outerplanar graphs). Now, since ∆(J) = 4, J has a proper 4-edge coloring. This proper edge coloring, together with the coloring of C is the required improper interval edge coloring of G.
Let s i = i j=1 n j (1 ≤ i ≤ r 2 ) and t i = ⌈ r 2 ⌉+i j=⌈ r 2 ⌉+1 n j (1 ≤ i ≤ r 2 ). Now let us consider the subgraphs H and H ′ of K n 1 ,n 2 ,...,nr induced by the vertices of X and Y , respectively. We first define an edge coloring α of H ∪ H ′ .
By the definition of α, we have Clearly, this upper bound on the impropriety in Theorem 4.17 is sharp for all Cartesian products of graphs when the factors are interval colorable. Let us note that there are graphs G and H such that µ int (G H) < max {µ int (G), µ int (H)}. For example, if G and H are both isomorphic to the Petersen graph, then, by Proposition 3.1, µ int (G) = µ int (H) = 2, but µ int (G H) = 1, since G and H contain perfect matchings [23]. On the other hand, if we consider the Cartesian product of two odd cycles C 2m+1 C 2n+1 , then again, by Proposition 3.1, µ int (C 2m+1 ) = µ int (C 2n+1 ) = 2, but in this case µ int (C 2m+1 C 2n+1 ) = 2, since C 2m+1 C 2n+1 is Class 2. So, the upper bound on the impropriety in Theorem 4.17 is also sharp for all Cartesian products of regular graphs when the factors and the Cartesian product of factors are Class 2.

The number of colors in an improper interval coloring
Following [18], we denote byt(G) the maximum number of colors used in an improper interval edge coloring of G. In [18], the authors proved the following two results.
Moreover, the upper bound is sharp.
Here we slightly improve the general upper bound from the last theorem. Clearly, H is a bipartite graph with |V (H)| = 2n.
We note that the upper bound in the preceding theorem is sharp by the example of a complete graph with only two vertices.