Bounding the monomial index and ( 1 , l )-weight choosability of a graph Ben Seamone

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Bounding the monomial index and (1,l)-weight choosability of a graph Ben Seamone


Introduction
A graph G = (V, E) will be simple and loopless unless otherwise stated.An edge k-weighting, w, of G is an assignment of a number from [k] := {1, 2, . . ., k} to each e ∈ E(G), that is w : Karoński, Łuczak, and Thomason [5] conjecture that, for every graph without a component isomorphic to K 2 , there is an edge 3-weighting such that the function S : V (G) → Z given by S(v) = e v w(e) is a proper colouring of V (G) (in other words, any two adjacent vertices have different sums of incident edge weights).If such a proper colouring S exists, then w is a vertex colouring by sums.Let χ e Σ (G) be the smallest value of k such that a graph G has an edge k-weighting which is a vertex colouring by sums.A graph G is nice if it contains no component isomorphic to K 2 .Karoński, Łuczak, and Thomason's conjecture (frequently called the "1-2-3 Conjecture") may be expressed as follows: 1-2-3 Conjecture.If G is a nice graph, then χ e Σ (G) ≤ 3. The best known upper bound on χ e Σ (G) is due to Kalkowski, Karoński and Pfender [4], who show that χ e Σ (G) ≤ 5 if G is nice.
In [2], Bartnicki, Grytczuk and Niwczyk consider a list variation of the 1-2-3 Conjecture.Assign to each edge e ∈ E(G) a list of k real numbers, say L e , and choose a weight w(e) ∈ L e for each e ∈ E(G).The resulting function w : E(G) → ∪ e∈E(G) L e is called an edge k-list-weighting.Given a graph G, the smallest k such that any assignment of lists of size k to E(G) permits an edge k-list-weighting which is a vertex colouring by sums is denoted ch e Σ (G) and called the edge weight choosability number of G.The following, stronger, conjecture is proposed in [2]: List 1-2-3 Conjecture.If G is a nice graph, then ch e Σ (G) ≤ 3. A similar problem to the List 1-2-3 Conjecture for graphs is solved for digraphs in [2], where a constructive method is used to show that ch e Σ (D) ≤ 2 for any digraph D. An alternate proof which uses Alon's Combinatorial Nullstellensatz [1] may be found in [6].
Another variant of the 1-2-3 Conjecture allows each vertex v ∈ V (G) to receive a weight w(v); the colour of v is then w(v) + e v w(e) rather than e v w(e).Such a function w : V ∪ E → [k] is called a total k-weighting.The smallest k for which G has a total k-weighting that is a proper colouring by sums is denoted χ t Σ (G).A similar list generalization as above may be considered; the smallest k such that the list version holds is denoted ch t Σ (G).The following two conjectures are posed in [9] and [10,14] respectively: Though the 1-2 Conjecture remains open, Kalkowski [3] has shown that a total weighting w of G which properly colours V (G) by sums always exists with w(v) ∈ {1, 2} and w(e) ∈ {1, 2, 3} for all v ∈ V (G), e ∈ E(G).
In [14], Wong and Zhu study (k, l)-total list-assignments, which are assignments of lists of size k to the vertices of a graph and lists of size l to the edges.If any (k, l)-total list-assignment of G permits a total weighting which is a vertex colouring by sums, then G is (k, l)-weight choosable.Obviously, if a graph G is (k, l)-weight choosable, then ch t Σ (G) ≤ max{k, l}.The List 1-2 Conjecture is equivalent to the statement that every graph is (2, 2)-choosable.Wong and Zhu [14] further conjecture that every nice graph is (1, 3)-weight choosable, a strengthening of the List 1-2-3 Conjecture.A recent breakthrough by Wong and Zhu [13] shows that every graph is (2, 3)-weight choosable and hence ch t Σ (G) ≤ 3 for every graph G.There is a good deal of literature on graph classes which are (k, l)-weight choosable for small values of k and l (see [2,8,10,11,12,14]).Of particular note, it is shown in [8] that every nice d-degenerate graph is (1, 2d)-weight choosable.However, whether or not there exists a constant l such that every nice graph is (1, l)-weight choosable remains an open question.
The purpose of this paper is show how the methods used in [2] can be extended to obtain the following general result.Let G = (V, E) be a nice graph, with E(G) = {e 1 , . . ., e m } and V (G) = {v 1 , . . ., v n }.Let ∂ 2 (G) denote the smallest value s such that every induced subgraph of G has vertices at distance 2 whose degrees sum to at most s.Associate with each e i the variable x i , with each v j the variable x m+j , and let X vj = ei vj x i for each v j .We show that the polynomial (u,v)∈E(D) (X v − X u ) has a monomial term in its expansion with non-zero coefficient for which no exponent exceeds ∂ 2 (G).Based on the work of [2], this implies that every nice graph is (1, ∂ 2 (G) + 1)-weight choosable.
The structure of the paper is as follows.In Section 2, we present a Combinatorial Nullstellensatz approach to the List 1-2-3 and List 1-2 Conjectures, which relies on calculating permanents of matrices which arise from natural colouring polynomials (one of which is given in the previous paragraph).Section 3 contains some intermediary lemmas on matrix permanents and colouring polynomials.Sections 2 and 3 are largely reliant on the results found in [2,10]; results are presented in near full detail, with examples, in the interest of keeping the article self-contained.Some results are generalized where necessary.Section 4 contains the main result of this paper, given above.The result that every nice graph is (1, ∂ 2 (G) + 1)-weight choosable is, unfortunately, weaker than that of Pan and Yang [8], however we are able to obtain improved bounds for some graph products in Section 5.

The permanent method and Alon's Nullstellensatz
Let G = (V, E) be a graph, with E(G) = {e 1 , . . ., e m } and V (G) = {v 1 , . . ., v n }.Associate with each e i the variable x i and with each v j the variable x m+j .Define two more variables for each v j ∈ V (G): X vj = ei vj x i and Y vj = x m+j + X vj .For an orientation D of G, define the following two polynomials, where l = m + n: Let w be an edge weighting of G.By letting x i = w(e i ) for 1 ≤ i ≤ m, w is a proper vertex colouring by sums if and only if P D (w(e 1 ), . . ., w(e m )) = 0.A similar conclusion can be made about T D if w is a total weighting of G.This leads us to the problem of determining when the polynomials P D and T D do not vanish everywhere, i.e., when there exist values of the variables for which the polynomial is non-zero.Alon's famed Combinatorial Nullstellensatz gives sufficient conditions to guarantee that a polynomial does not vanish everywhere.
Combinatorial Nullstellensatz (Alon [1]).Let F be an arbitrary field, and let f = f (x 1 , . . ., x n ) be a polynomial in F[x 1 , . . ., x n ].Suppose the total degree of f is n i=1 t i , where each t i is a nonnegative integer, and suppose the coefficient of For a polynomial P ∈ F[x 1 , . . ., x l ] and a monomial term M of P , let h(M ) be the largest exponent of any variable in M .The monomial index of P , denoted mind(P ), is the minimum h(M ) taken over all monomials of P .Define the graph parameters mind(G) := mind(P D ) and tmind(G) := mind(T D ), where D is an orientation of G.Note that, given a graph G and two orientations D and D , P D (x 1 , . . ., x l ) = ±P D (x 1 , . . ., x l ); a similar argument holds for T D .The parameters mind(G) and tmind(G) are hence well-defined.Note that, for any graph G, tmind(G) ≤ mind(G).
The following lemma is obtained by applying the Combinatorial Nullstellensatz to P D and T D : Lemma 2.1.Let G be a graph and k a positive integer.
2. (Przybyło, Woźniak [10]) If tmind(G) ≤ k, then ch t Σ (G) ≤ k + 1.More generally, Wong and Zhu show the following: Lemma 2.2 (Wong, Zhu [14]).Let G be a nice graph, D an orientation of G, and This leads to the following simple corollary: The following proposition allows us to consider only connected graphs.
In [2], Bartnicki et al. show how one may study the permanent of particular {−1, 0, 1}-matrices in order to gain insight on mind(G) and tmind(G).Let M(m, n) denote the set of all real valued matrices with m rows and n columns, and M(m) denote the set of square m × m matrices.The permanent of a matrix A ∈ M(m), denoted per A, is calculated as follows: The permanent may also be defined for a general matrix A ∈ M(m, n) if n ≥ m.Let Q m,n denote the set of sequences of length m with entries from [n] which contain no repetition of elements; such sequences are also known as m-permutations from [n].For example, Q 2,3 = {(1, 2), (1, 3), (2, 1), (2, 3), (3, 1), (3, 2)}.The permanent of A is defined as follows: The permanent rank of a matrix A (not necessarily square) is the size of the largest square submatrix of A having nonzero permanent.Let Alternately, pind(A) is the smallest k such that a square matrix of size m having nonzero permanent can be constructed by taking columns from A, each column taken no more than k times.
There are three matrices related to directed graphs which will be of interest: • A D = (a i,j ) where a i,j =    1 if e j is incident with the head of e i −1 if e j is incident with the tail of e i 0 otherwise The following lemmas, which relate the matrices A D , B D , and M D to the polynomials P D and T D , provide the fundamental link between the graphic polynomials of interest and matrix permanents: The proof is omitted, but the result follows from the fact that the coefficient of where M is the m × m matrix where column a j from A appears k j times.Lemma 2.6 immediately implies the following vital link between the (total) monomial index of a graph G and the permanent index of A D (respectively, T D ) for any orientation D of G: Lemma 2.7.Let D be an orientation of a graph G. [10]) For any graph G, tmind(G) = pind(M D ).

(Przybyło, Woźniak
Lemmas 2.1 and 2.7 imply that ch e Σ (G) ≤ pind(A D ) + 1 (if G is nice) and ch t Σ (G) ≤ pind(M D ) + 1.We note the following general result of Wong and Zhu [14].Let G be a graph, D an orientation of G, and M D the matrix defined above.Suppose a matrix M ∈ M(m) has only columns taken from M D and has per (M ) = 0.If no column associated with an edge e ∈ E(G) appears more than l times and no column associated with a vertex v ∈ V (G) appears more than k times, then G is (k + 1, l + 1)-weight choosable.However, in light of the theorem in [13] stating that every graph is (2, 3)-weight choosable, it is now sufficient to consider only (1, l)-weight choosability (and hence to bounding mind(G)) and (2, 2)-weight choosability (and hence trying to show that tmind(G) = pind(M D ) = 1).
In summary, to determine upper bounds on ch e Σ (G), ch t Σ (G) or values of k and l for which G is (k, l)weight choosable, it is sufficient to consider the permanents of matrices obtained by replicating columns from M D for some orientation D of G. Consider the following illustrative example.Let D be the digraph in Figure 1 and let G be its underlying simple graph.
The associated polynomial, P D , is e 3 e 4 e 5 e 6 Fig. 1: A digraph used to illustrate AD, BD, and MD Note that for each factor f of P D , the coefficients of x 1 , . . ., x 6 are equal to the entries appearing on the row of A D corresponding to f .We have: Since per A D = −4 = 0, we have pind(A D ) = 1 (each column from A D is chosen once).By Lemma 2.7(1), mind(G) = 1, and so G is (1, 2)-weight choosable by Corollary 2.3.Note that no graph is (1, 1)-weight choosable, so l = 2 is the minimum value for which G is (1, l)-weight choosable.We also clearly have that ch e Σ (G) ≤ 2. Since there are adjacent vertices of equal degree in G, we have χ e Σ (G) = 1, implying ch e Σ (G) = 1 and so ch e Σ (G) = 2.

Some intermediary results on permanent indices and monomial indices
The major results of this paper are proven by establishing bounds on mind(G) using the permanent method outlined in the previous section.One important tool is the following lemma, a generalization of a similar result in [2]: Lemma 3.1 (Przybyło, Woźniak [10]).Let A be an m×l matrix, and let L be an m×m matrix where each column of L is a linear combination of columns of A. Let n j denote the number of columns of L in which the j th column of A appears with nonzero coefficient.If per L = 0, then pind(A) ≤ max {n j | j = 1, . . .l}.
We will also find the following theorem useful, which gives a method for constructing graphs in a way that preserves the property of having low monomial index: Theorem 3.2 (Bartnicki, Grytczuk, Niwczyk [2]).Let G be a simple graph with mind(G) ≤ 2. Let U be a nonempty subset of V (G).If F is a graph obtained by adding two new vertices u, v to V (G) and joining them to each vertex of U , and H is a graph obtained from F by joining u and v, then mind(F ), mind(H) ≤ 2.
As a consequence, the following graph classes have low monomial index and hence small values of ch e Σ (G) by Corollary 2.1(1): ).If G is a complete graph, a complete bipartite graph, or tree, then mind(G) ≤ 2.
Consider the colouring polynomial Since each variable appears in exactly two factors of P D , no exponent in the expansion of P D exceeds 2, and hence mind(G) ≤ 2.
In order to prove our major results in Section 4, the following generalization of Theorem 3.2 is required: Let G be a graph with finite monomial index mind(G) ≥ 1.Let U be a nonempty subset of V (G).If F is a graph obtained by adding two new vertices u, v to V (G) and joining them to each vertex of U , and F * is a graph obtained from F by joining u and v, then mind(F ), mind(F * ) ≤ max{2, mind(G)}.
The proof which follows is an adaptation of the proof of Theorem 3.2 found in [2].Given a matrix A with columns a 1 , a 2 , . . ., a n and a sequence of (not necessarily distinct) column indices K = (i 1 , i 2 , . . ., i k ), A(K) is defined to be the matrix Proof: Let U = {u 1 , . . ., u k } be the subset of V (G) stated in the theorem.Let E u = {e 1 , e 3 , . . ., e 2k−1 } and E v = {e 2 , e 4 , . . ., e 2k } be the sets of edges incident to the vertices u and v, respectively.Assume that these edges are oriented toward U , and that for each i = 1, 2, . . ., k the edges e 2i−1 and e 2i have the same head.
Let D be an orientation of F , D the induced orientation of G, and consider the matrices A D and A D .Let A 1 , . . ., A 2k be the first 2k columns of A D , corresponding to {e 1 , e 2 , . . ., e 2k }.If we write Since the edges e 2i−1 and e 2i have the same head for each i = 1, 2, . . ., k, the columns A 2i−1 and A 2i agree on Z. Furthermore, Y may be written as a block matrix, where ( 0 1 1 0 ) occupies the diagonals and −1 0 0 −1 is everywhere else, as seen in Figure 2.
There exists a matrix of columns from A D , with no column used more than mind(G) times, with nonzero permanent.Let K denote the sequence of edges of G which index this matrix.Consider a new matrix The properties of the columns of A outlined above imply that the matrix M can be written as follows: , where R has all constant rows: .
Since per M = per R × per A D (K), each of per R and per A G (K) are nonzero, and any column of A appears in the linear combination of at most 2 columns of M , Lemma 3.1 implies that mind(F ) ≤ max{2, mind(G)}.
We now consider F * .Let H be an orientation of F * with e 0 = uv oriented from v to u.The matrix A H is precisely A D with a row and column added for e 0 (say, as the first row and column).It can be depicted in block form A H = Y X Z A G , where Y and Z are the matrices depicted in Figure 3.
where R is the following square matrix: .
It is shown in [2] that per R = 0. Hence per N = per R × per A G (K) = 0, and since any column of A appears in the linear combination of at most 2 columns of N , Lemma 3.1 implies thats mind(H) ≤ max{2, mind(G)}.

A general bound for a graph's monomial index
Armed with the Combinatorial Nullstellensatz and the permanent method, we may now proceed with our main results.
Recall that a graph G is d-degenerate if every induced subgraph of G has a vertex of degree at most d.If G and H are graphs, we write H ≤ i G to denote that H is an induced subgraph of G.The degeneracy of a graph G, which we denote by ∂(G), is the smallest We extend the notion of degeneracy to pairs of vertices at a given distance.Given an integer r ≥ 1, let δ r (G) denote the minimum value of d(u) If no induced subgraph of G has vertices at distance exactly r (for example, G = K n and r ≥ 2), then we adopt the convention that ∂ r (G) = 2∆(G).
We now show that mind(G) is at most ∂ 2 (G).The result is achieved by carefully orienting the edges of a graph and applying the lemmas from the previous sections to show that our desired matrix has non-zero permanent.
Proof: We may assume that G is connected, since Proposition 2.4 states that mind(G) is at most the largest monomial index of its components.If G is a tree, cycle, or complete graph, then mind(G) ≤ 2 by Corollary 3.3 and Proposition 3.4, and hence the theorem holds for the following graphs: P 3 , K 3 , P 4 , K 1,3 , C 4 , and K 4 .If G is isomorphic to K 3 with a leaf or C 4 with a chord, then one may check that the theorem holds for G by straightforward computation of the associated colouring polynomial P D for any orientation D. Hence, the theorem holds for any connected graph on 3 or 4 vertices.
We proceed now by induction on |V (G)|.Let G be a connected graph on at least 5 vertices, and for any graph If G is a complete graph, then the theorem holds by Corollary 3.3.Assume that G is not complete.There exist u, v, w ∈ V (G) such that the induced subgraph G[{u, v, w}] is a path of length 2 (or, uvw is an induced 2-path).Choose this 2-path such that d(u) + d(w) is minimum (and, hence, The ultimate goal will be to apply an inductive argument to G − {u, w}, however we must concern ourselves with whether or not this subgraph of G is nice.To this end, we define the following sets of edges: The path uvw and the sets of edges E u , E v , E w are shown in Figure 4.If x is adjacent to both u and w, then v and x are adjacent twins.Suppose that G \ {v, x} is not nice; we will show that this contradicts our choice of uvw which minimizes If G is not nice, then u, w, or both u and w are adjacent to exactly one vertex in G other than v and x; without loss of generality, suppose that uy ∈ E(G), y = v, x.Since y / ∈ N G (x), the vertices y, u, x induce a 2-path; furthermore, d G (y) + d G (x) = 1 + 3 = 4.This contracts our choice of uvw, since d(u) + d(w) ≥ 3 + 2 = 5.Thus, G − {v, x} is a nice graph, and so, by Lemma 3.5, mind(G) ≤ max{2, mind(G − {v, x})}.By the induction hypothesis, mind We may now assume that x is not adjacent to at least one of u and w.If w / ∈ N G (x), then both uvw and xvw are induced 2-paths in G.By the minimality of d(u) + d(w), we must have that d(u) ≤ d(x).If u is adjacent to x, then d(x) = 2 and, since u is adjacent to v as well, and N G (u) = {v}.In either case, u and x are twins.If G − {u, x} is not nice, then the only edge not incident to u or x is the edge vw, contradicting our choice of G with |V (G)| ≥ 5. Assume that G − {u, x} is nice.By Lemma 3.5, mind(G) ≤ max{2, mind(G − {u, x})}, and by the induction hypothesis, mind

If u /
∈ N G (x) and w ∈ N G (x), then the exact same argument holds as for u ∈ N G (x) and w / ∈ N G (x). Having considered all possible neighbourhoods of x, we conclude that if Suppose that E v ∩F = ∅.The argument proceeds as follows: after choosing a "good" orientation D of G, we will construct a matrix whose columns are linear combinations of A D with no column of A D being used more than ∂ 2 (G) times and with nonzero permanent.The result then follows by Lemma 3.1.
Let D be an orientation of G where the edges of E u ∪ {uv} and E v are oriented toward u and v, respectively, and the edges of E w ∪ {vw} are oriented away from w; see Figure 5.Let We must still concern ourselves with the possibility that deleting u and w from G gives a graph which is not nice.If a component of G − {u, w} is isomorphic to K 2 , then one vertex of this component must be adjacent to either u or w in G. Let F = {f 1 , . . ., f k } be the set of edges belonging to the k connected components of G − {u, w} that are isomorphic to K 2 .For each f i ∈ F, let e i be an edge from E u or E w to which f i is adjacent.Let F denote this collection of edges from E u ∪ E w , and let F u = {e : e ∈ E u ∩ F } and F w = {e : e ∈ E w ∩ F }.Each edge f i ∈ F will be oriented away from its shared endpoint with e i .
Let H = G − {u, w} − F and D(H) be the corresponding sub-digraph of D. Since we have removed all components isomorphic to K 2 , H is nice.Since H has fewer vertices than G, by Fig. 6: An operation on two columns of AD the induction hypothesis, mind(H) ≤ ∂ 2 (H).Hence, there exists a matrix L H consisting of columns of A D(H) , none repeated more than ∂ 2 (H) times, with per (L H ) = 0. Let K denote the sequence of edges which indexes the columns of L H .For an m × n matrix A, recall that A (k) is the m × kn matrix consisting of k consecutive copies of A (see page 176).Let L G be the following block matrix: where the blocks are as follows: • J d(u)+d(w) is the (d(u) + d(w)) × (d(u) + d(w)) all 1's matrix.
• K = K1 K2 having entries depending on whether the column is indexed by e i ∈ F u or e i ∈ F w .If the column is indexed by e i ∈ F u , then the column will have (i) 1 in each row indexed by the other edges from E u , (ii) 1 in the row indexed by uv, (iii) −1 in the row indexed by f i , and (iv) 0 in all other entries.Otherwise, the entries follow the same pattern with the signs swapped.Since the column associated with e i has only one non-zero entry in the rows indexed by F, K 2 is diagonal with |F u | entries being −1 and |F w | entries being 1.
• L H , is the matrix with per (L H ) = 0 defined above.
Since J d(u)+d(w) , K 2 , and L H are all square matrices, per We immediately obtain the following result by Corollary 2.3: and so Corollary 4.2 does not improve upon the constructive result of Pan and Yang [8] that every nice d-degenerate graph is (1, 2d)-weight choosable.However, we hope that the extension of the algebraic methods established by Bartnicki et al given in this section will serve as motivation for subsequent improvements.In particular, the proof relied on finding a "good" induced subgraph whose columns had cancellation properties that could be exploited in calculating the permanent index of the matrix A D .It is conceivable that a more clever choice of induced subgraph might yield a better result than that given in Theorem 4.1.

Monomial indices of graph products
We now consider some classes of graphs where we can improve upon the result on Theorem 4.1, in particular the cartesian product of two graphs.The following decomposition lemma on mind(G) provides an approach for such graphs: Lemma 5.1.Let G be a graph, and let H be an induced subgraph of G containing a 2-factor.Let X be a minimal edge cut separating Proof: Let |V (H)| = v and F = {e 1 , . . ., e v } be a 2-factor of H. Let D be an orientation of G such that the cycles of F are directed.Define the column vector c = v i=1 c i where c i is the column of A D corresponding to e i .For each e ∈ E(H) \ F there are two edges of F incident to each of the head and tail of e, and for each e ∈ F there is one edge of F incident to each of the head and tail of e.Hence, the entries of c are nonzero in the rows indexed by the edges of X and 0 in all other entries.
There exists a matrix L G−X consisting of columns of A G−X with no column of A D repeated more than mind(G − X) times and per (L G−X ) = 0. Let K denote the sequence of edges of G − X which index A G−X .Consider the following matrix: where (M N ) is indexed by X, each row of M is constant, and every entry of M is nonzero.Any column indexed by e ∈ E(G) \ F is used at most mind(G − X) times in the construction of L, and any edge from Recall that the Cartesian product of two graphs G and H, denoted by G 2 H, is defined as the graph having vertex set V (G) × V (H) where two vertices (u, u ) and (v, v ) are adjacent if and only if either u = v and u is adjacent to v in H or u = v and u is adjacent to v in G.Some results on χ e Σ (G) for Cartesian products of graphs are given in [7]  Recall that Pan and Yang [8] showed that every nice d-degenerate graph is (1, 2d)-weight choosable.Since G 2 K n is d(n − 1) degenerate, where d is the degeneracy of G, Corollary 5.4 represents an improvement for these graphs.
For an orientation D of G, define the matrices A D ∈ M(m), B D ∈ M(m, n), and M D ∈ M(m, m + n) as follows:

Fig. 3 :
Fig. 3: The matrices Y and Z Let A 0 , A 1 , . . ., A 2k denote the first 2k + 1 columns of A H , corresponding to the edges e 0 , e 1 , . . ., e 2k .Form a new matrix N= (N 0 N 0 N 1 N 2 N 2 • • • N k N k B(K)) , then there can be only one edge in this intersection, otherwise the connected component containing v in G − {u, w} would have two or more edges.Since uv, vw ∈ E(G), it follows that N G (v) = {u, w, for some vertex x ∈ V (G).Since {v, x} induces a graph isomorphic to K 2 in G − {u, w}, we have that N G (x) ⊆ {u, v, w}.

Fig. 5 :
Fig. 5: An orientation D of a graph G with an induced 2-path uvw c uv and c vw be the columns of A D associated with the edges uv and vw, respectively, and let c = c uv − c vw ; see Figure 6.
; for instance, if G and H are regular and bipartite, then χ e Σ (K n 2 H), χ e Σ (C t 2 H), and χ e Σ (G 2 H) are at most 2 for n ≥ 4, and t ≥ 4, t = 5.Lemma 5.1 may be used to bound ch e Σ (G 2 H) for many more graphs G and H.Note that, for the graph G 2 H and vertex v ∈ V (G), the subgraph induced by the set of vertices {(v, x) : x ∈ V (H)} is denoted (v, H).Theorem 5.2.Let H be a regular graph on n ≥ 3 vertices which contains a 2-factor.If G is a ddegenerate graph, then mind(G 2 H) ≤ nd + mind(H).