Hoffman’s theorem on feasible circulations and Ghouila-Houry’s theorem on feasible tensions are classical results of graph theory. Camion generalized these results to systems of inequalities over regular chain groups. An analogue of Camion’s result is proved in which solutions can be forced to be odd valued. The obtained result also generalizes the results of Pretzel and Youngs as well as Slilaty. It is also shown how Ghouila-Houry’s result can be used to give a new proof of the graph- coloring theorem of Minty and Vitaver

Daniel Slilaty

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Discrete Mathematics Letterswww.dmlett.comDiscrete Math. Lett. 12 (2023) 62–65DOI: 10.47443/dml.2023.033Research ArticleOdd solutions to systems of inequalities coming from regular chain groupsDaniel Slilaty∗Department of Mathematics and Statistics, Wright State University, Dayton, Ohio, USA(Received: 27 February 2023. Received in revised form: 2 June 2023. Accepted: 5 June 2023. Published online: 20 June 2023.)c© 2023 the author. This is an open-access article under the CC BY (International 4.0) license (www.creativecommons.org/licenses/by/4.0/).AbstractHoffman’s theorem on feasible circulations and Ghouila-Houry’s theorem on feasible tensions are classical results of graphtheory. Camion generalized these results to systems of inequalities over regular chain groups. An analogue of Camion’sresult is proved in which solutions can be forced to be odd valued. The obtained result also generalizes the results of Pretzeland Youngs as well as Slilaty. It is also shown how Ghouila-Houry’s result can be used to give a new proof of the graph-coloring theorem of Minty and Vitaver.Keywords: unimodular chain group; regular chain group; totally unimodular matrix; systems of inequalities; graphcoloring.2020 Mathematics Subject Classification: 05C15, 05A20, 05C50.1. IntroductionHoffman’s theorem on feasible circulations [5] and Ghouila-Houry’s theorem on feasible tensions [4] are classical resultsof graph theory. Within the ring of integers, Camion [2] generalized both of these results using the setting of regularchain groups (also known as unimodular modules).† Within the more general algebraic framework developed by Ghouila-Houry [5], Camion [3] again generalized both results. In this paper we will only be working within the ring of integers.Our idea might extend to the more general setting of [3,4] by using an index-2 subgroup of the interval group.Let E be a finite set. A Z-chain group is a submodule C of ZE . The elements of C are called chains. A chain c ∈ C maybe thought of as a function c : E → Z. The support of c is |c| = {e ∈ E : c(e) 6= 0}; furthermore, |c| naturally partitionsinto subsets |c|+ = {e ∈ E : c(e) > 0} and |c|− = {e ∈ E : c(e) < 0}. A chain c ∈ C is elementary when |c| is nonempty andsubset minimal among all non-zero chains in C. A chain c ∈ C is primitive when c(e) ∈ {−1, 0, 1} for all e. A Z-chain groupC is a regular chain group (also called a unimodular module) when every elementary chain satisfies c = kc′ where c′ ∈ C isprimitive and k ∈ Z. It is known that the nullspace of a totally unimodular matrix is a regular chain group and a regularchain group is the nullspace of some totally unimodular matrix. The orthogonal complement of a regular chain group isalso a regular chain group. The integer cycle and cocycle spaces of a graph G are orthogonal complements of each otherand are both regular chain groups.‡Let I be an assignment of closed Z-intervals to each e ∈ E; that is, for each e ∈ E, I(e) = [l(e), u(e)] is a closed intervalin Z (possibly of length 0). Extend I to each c ∈ C by settingl(c) =∑e∈|c|+c(e)l(e) +∑e∈|c|−c(e)u(e) and u(c) =∑e∈|c|+c(e)u(e) +∑e∈|c|−c(e)l(e).For primitive chains this simplifies tol(c) =∑e∈|c|+l(e)−∑e∈|c|−u(e) and u(c) =∑e∈|c|+u(e)−∑e∈|c|−l(e).Given any chain d ∈ ZE , there is an associated homomorphism d̂ : C → Z defined by d̂(c) =∑e∈|c| d(e)c(e). Thus d̂ = 0if and only if d ∈ C∗ and d̂1 = d̂2 if and only if d1 − d2 ∈ C∗.∗E-mail address: daniel.slilaty@wright.edu†Seymour proved a special case of Camion’s result in [9, (14.5)] which he cites as a previously known folklore result.‡Seymour famously proved [8] that regular chain groups decompose along small separations into cycle spaces of graphs, cocycle spaces of graphs, andcopies of the chain group associated with the matroid R10.D. Slilaty / Discrete Math. Lett. 12 (2023) 62–65 63Camion’s result may be stated as follows.Theorem 1.1 (Camion [2]). Let C ≤ ZE be a regular chain group, let f : C → Z be a homomorphism, and let I be anassignment Z-intervals to E. The following are equivalent.(1) f(c) ∈ I(c) for each primitive chain c ∈ C.(2) f(c) ∈ I(c) for each chain c ∈ C.(3) There is d ∈ ZE satisfying d̂ = f and d(e) ∈ I(e) for each e ∈ |d|.Define the length of c ∈ ZE as length(c) =∑e∈|c| |c(e)|. This, of course, generalizes the notion of the length of a cyclein a graph. Theorem 1.2 is a “parity-respecting” version of Theorem 1.1. When C is the integer cycle space of a graphand all intervals are [−k, k] for some odd k, Theorem 1.2 specializes to a result of Slilaty [10]. For k = 1, Slilaty’s resultspecializes to a result of Pretzel and Youngs [7]. A notable consequence of this theorem is the chain d obtained in Part (3)is everywhere non-zero.Theorem 1.2. Let C ≤ ZE be a regular chain group, let f : C → Z be a homomorphism for which f(c) ≡ length(c) (mod 2)for each c ∈ C, and let I be an assignment Z-intervals to E for which both u(e) and l(e) are odd for each e ∈ E. The followingare equivalent.(1) f(c) ∈ I(c) for each primitive chain c ∈ C.(2) f(c) ∈ I(c) for each chain c ∈ C.(3) There is d ∈ ZE satisfying d̂ = f and d(e) ∈ I(e) with d(e) odd for each e ∈ |d|.Theorem 1.2 will be proven in Section 2. In relation to the discussion in this paper, it is relevant to note that Ghouila-Houry’s Theorem (or Theorem 1.1) can be used to give a new proof in the non-trivial direction of the elegant Minty-VitaverTheorem [6,12] stated below. We will give this proof in Section 3. An acyclic orientation f of a graph G partitions the edgesof any cycle C in G into two nonempty sets, C+ and C−. Letratio(C, f) = max{|C+||C−|,|C−||C+|}.Theorem 1.3 (Minty [6] and Vitaver [12].). A graph G is k-colorable if and only if there is an acyclic orientation f of G forwhich ratio(C, f) ≤ k− 1 for all cycles C in G.2. Proof of Theorem 1.2Chain c′ conforms to chain c when |c′| ⊆ |c| and c′(e)c(e) ≥ 0 for all e ∈ E. Tutte proved [11, (5.43)] that every c in a regularchain group C has a decomposition c = c1 + · · · + cm in which each ci ∈ C is primitive and conforms to c. Our proof ofTheorem 1.2 is much like Seymour’s proof in [9, (14.5)] with the added concern of keeping track of parities.Proof of Theorem 1.2. (1→ 2) Any c ∈ C decomposes as c = c1+ · · ·+cm in which each ci is primitive and conforms to c. Nowbecause each ci conforms to c and satisfies f(ci) ∈ I(ci) we may add all of the associated inequalities to obtain f(c) ∈ I(c).(2→ 1) Follows trivially from the fact that primitive chains are chains.(3→ 1) Suppose that d ∈ ZE satisfies (3). Then, for any primitive c ∈ Cf(c) =∑e∈Ec(e)d(e) =∑e∈|c|+d(e)−∑e∈|c|−d(e) ≤ u(c).Similarly, f(c) ≥ l(c) and so f(c) ∈ I(c), as required.(1 → 3) Suppose that f(c) ∈ I(c) for all c ∈ C. We proceed by induction on the quantity L(I) =∑e∈E(u(e) − l(e)). IfL(I) = 0, then u(e) = l(e) for all e and so f(c) ∈ I(c) implies that I(c) = [f(c), f(c)] for each c. Thus∑e∈Eu(e)c(e) = f(c),which along with the fact that u(e) = l(e) is odd for each e ∈ E makes u = l our desired element of ZE .63D. Slilaty / Discrete Math. Lett. 12 (2023) 62–65 64Now take interval assignment I such that L(I) > 0. Thus there is e0 ∈ E for which u(e0) − l(e0) > 0. Since l(e0) andu(e0) are both odd, u(e0)− l(e0) ≥ 2. Let I ′ be an interval assignment such that I ′(e) = I(e) for all e 6= e0 andI ′(e0) = [l(e0), u(e0)− 2].Also, let I ′′ be an interval assignment such that I ′′(e) = I(e) for all e 6= e0 and I ′′(e0) = [l(e0) + 2, u(e0)]. EvidentlyL(I ′) = L(I ′′) = L(I)− 2 and for all e ∈ E the intervals I ′(e) and I ′′(e) have odd-valued endpoints. So, if either f(c) ∈ I ′(c)for all primitive c ∈ C or f(c) ∈ I ′′(c) for all primitive c ∈ C, then by induction we get d ∈ ZE satisfying (3). If neither of theseconditions hold, then there are primitive chains c1, c2 ∈ C with e0 ∈ |c1|+ ∩ |c2|+ such that f(c1) /∈ I ′(c1) and f(c2) /∈ I ′′(c2).Because f(c) ≡ length(c) (mod 2) for all c ∈ C and l(e) and u(e) are odd for each e ∈ E,f(c) ≡ length(c) ≡ l(c) ≡ u(c) (mod 2)for all c ∈ C. So now because f(c1) ∈ I(c1) and f(c2) ∈ I(c2) we get thatf(c1) = u(c1) =∑e∈|c1|+u(e)−∑e∈|c1|−l(e) andf(c2) = l(c2) =∑e∈|c2|+l(e)−∑e∈|c2|−u(e).Replace c2 with −c2 to obtainf(c2) = u(c2) =∑e∈|c2|+u(e)−∑e∈|c2|−l(e)where we now have e0 ∈ |c1|+ ∩ |c2|−. Now let z = c1 + c2. Since we have already proven 1 ←→ 2 we get the following inwhich A = (|c1|+ ∩ |c2|−) ∪ (|c1|− ∩ |c2|+):f(z) ≤ u(z)=∑e∈|z|+z(e)u(e) +∑e∈|z|−z(e)l(e)=∑e∈|c1|+u(e)−∑e∈|c1|−l(e) +∑e∈|c2|+u(e)−∑e∈|c2|−l(e)−∑e∈A(u(e)− l(e))= f(c1) + f(c2)−∑e∈A(u(e)− l(e))= f(z)−∑e∈A(u(e)− l(e)).Now because e0 ∈ A we obtain 0 ≤ −∑e∈A(u(e)− l(e)) ≤ −2, a contradiction.3. A new proof of the Minty-Vitaver TheoremSay that f is an acyclic orientation of G for which ratio(C, f) ≤ k− 1 for all cycles C. Let C be the integer cycle space of thedirected graph (G, f) and assign to each e the interval [1, k − 1]. Now and let c ∈ C be any primitive and also elementarychain. (We will consider all primitive chains later.) Thus |c| corresponds to the edge set of a cycle, call it C, in G wherewithout loss of generality |c|+ = C+ and |c|− = C− with respect to f . By assumption we have that |C+|/|C−| and |C−|/|C+|are both at most k − 1. Thus we have thatl(c) =∑e∈|C+|1−∑e∈|C−|(k − 1) = |C+| − (k − 1)|C−| ≤ |C+| − |C+| = 0and also thatu(c) =∑e∈|C+|(k − 1)−∑e∈|C−|1 = (k − 1)|C+| − |C−| ≥ |C−| − |C−| = 0.Thus l(c) ≤ 0 ≤ u(c) for any primitive and elementary c ∈ C. Now if c is any primitive chain in C, then because c isorthogonal to all of the chains in the cocycle space C∗ (in particular, the vertex cocycles) |c| corresponds to a subgraph ofthe directed graph (G, f) in which each vertex has equal in-degree and out-degree. Thus |c| partitions into edge-disjointdirected cycles in (G, f), thus c = c1 + · · · + cn in which each ci is primitive, elementary, and conforms to c. So now since64D. Slilaty / Discrete Math. Lett. 12 (2023) 62–65 65we have proven that l(ci) ≤ 0 ≤ u(ci) for each ci, we now have that l(c) ≤ 0 ≤ u(c). Thus Ghouila-Houry’s Theorem (whichis a special case of Theorem 1.1 within the ring of integers) implies that there is d ∈ C∗ for which d(e) ∈ [1, k − 1] for eachedge e in G. A chain in C∗ is often called a tension in graph theory. Thus d (mod k) is a nowhere-zero Zk-tension. It is wellknown that, for a connected graph G, the proper k-colorings of G using color set Zk correspond k-to-1 to the nowhere-zeroZk-tensions of G. (See, for example, the discussion of potential functions in [1, Sec. 5.3].)References[1] C. Berge, Graphs and Hypergraphs, Revised Edition, North-Holland, Amsterdam, 1976.[2] P. Camion, Matrices Totalement Unimodulaires et Problmes Combinatoires, Ph.D. Thesis, Université Libre de Bruxelles, Brussels, 1963.[3] P. Camion, Unimodular modules, Discrete Math. 306 (2006) 2355–2382.[4] A. Ghouila-Houri, Sur l’existence d’un flot ou d’une tension prenant ses valeurs dans un groupe abélien, C. R. Acad. Sci. Paris 250 (1960) 3931–3933.[5] A. J. Hoffman, Some recent applications of the theory of linear inequalities to extremal combinatorial analysis, In: R. Bellman, M. Hall Jr. (Eds.), Combinatorial Analysis,Proc. Sympos. Appl. Math. 10, American Mathematical Society, Providence, 1960, 113–127.[6] G. J. Minty, A theorem on n-coloring the points of a linear graph, Amer. Math. Monthly 69 (1962) 623–624.[7] O. Pretzel, D. Youngs, Cycle lengths and graph orientations, SIAM J. Discrete Math. 3 (1990) 544–553.[8] P. D. Seymour, Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980) 305–359.[9] P. D. Seymour, Matroids and multicommodity flows, European J. Combin. 2 (1981) 257–290.[10] D. Slilaty, Integer functions on the cycle space and edges of a graph, Graphs Combin. 26 (2010) 293–299.[11] W. T. Tutte, Lectures on matroids, J. Res. Nat. Bur. Stand. Sect. B 69 (1965) 1–47.[12] L. M. Vitaver, Determination of minimal coloring of vertices of a graph by means of Boolean powers of the incidence matrix, Dokl. Akad. Nauk SSSR 147 (1962) 758–759.65

Odd solutions to systems of inequalities coming from regular chain groups

Slilaty, Daniel

CORE

Wright State University CORE Scholar Mathematics and Statistics Faculty Publications Mathematics and Statistics 2023 Odd Solutions to Systems of Inequalities Coming From Regular Chain Groups Daniel Slilaty Follow this and additional works at: https://corescholar.libraries.wright.edu/math  Part of the Applied Mathematics Commons, Applied Statistics Commons, and the Mathematics Commons This Article is brought to you for free and open access by the Mathematics and Statistics department at CORE Scholar. It has been accepted for inclusion in Mathematics and Statistics Faculty Publications by an authorized administrator of CORE Scholar. For more information, please contact library-corescholar@wright.edu. Discrete Mathematics Letterswww.dmlett.comDiscrete Math. Lett. 12 (2023) 62–65DOI: 10.47443/dml.2023.033Research ArticleOdd solutions to systems of inequalities coming from regular chain groupsDaniel Slilaty∗Department of Mathematics and Statistics, Wright State University, Dayton, Ohio, USA(Received: 27 February 2023. Received in revised form: 2 June 2023. Accepted: 5 June 2023. Published online: 20 June 2023.)c© 2023 the author. This is an open-access article under the CC BY (International 4.0) license (www.creativecommons.org/licenses/by/4.0/).AbstractHoffman’s theorem on feasible circulations and Ghouila-Houry’s theorem on feasible tensions are classical results of graphtheory. Camion generalized these results to systems of inequalities over regular chain groups. An analogue of Camion’sresult is proved in which solutions can be forced to be odd valued. The obtained result also generalizes the results of Pretzeland Youngs as well as Slilaty. It is also shown how Ghouila-Houry’s result can be used to give a new proof of the graph-coloring theorem of Minty and Vitaver.Keywords: unimodular chain group; regular chain group; totally unimodular matrix; systems of inequalities; graphcoloring.2020 Mathematics Subject Classification: 05C15, 05A20, 05C50.1. IntroductionHoffman’s theorem on feasible circulations [5] and Ghouila-Houry’s theorem on feasible tensions [4] are classical resultsof graph theory. Within the ring of integers, Camion [2] generalized both of these results using the setting of regularchain groups (also known as unimodular modules).† Within the more general algebraic framework developed by Ghouila-Houry [5], Camion [3] again generalized both results. In this paper we will only be working within the ring of integers.Our idea might extend to the more general setting of [3,4] by using an index-2 subgroup of the interval group.Let E be a finite set. A Z-chain group is a submodule C of ZE . The elements of C are called chains. A chain c ∈ C maybe thought of as a function c : E → Z. The support of c is |c| = {e ∈ E : c(e) 6= 0}; furthermore, |c| naturally partitionsinto subsets |c|+ = {e ∈ E : c(e) > 0} and |c|− = {e ∈ E : c(e) < 0}. A chain c ∈ C is elementary when |c| is nonempty andsubset minimal among all non-zero chains in C. A chain c ∈ C is primitive when c(e) ∈ {−1, 0, 1} for all e. A Z-chain groupC is a regular chain group (also called a unimodular module) when every elementary chain satisfies c = kc′ where c′ ∈ C isprimitive and k ∈ Z. It is known that the nullspace of a totally unimodular matrix is a regular chain group and a regularchain group is the nullspace of some totally unimodular matrix. The orthogonal complement of a regular chain group isalso a regular chain group. The integer cycle and cocycle spaces of a graph G are orthogonal complements of each otherand are both regular chain groups.‡Let I be an assignment of closed Z-intervals to each e ∈ E; that is, for each e ∈ E, I(e) = [l(e), u(e)] is a closed intervalin Z (possibly of length 0). Extend I to each c ∈ C by settingl(c) =∑e∈|c|+c(e)l(e) +∑e∈|c|−c(e)u(e) and u(c) =∑e∈|c|+c(e)u(e) +∑e∈|c|−c(e)l(e).For primitive chains this simplifies tol(c) =∑e∈|c|+l(e)−∑e∈|c|−u(e) and u(c) =∑e∈|c|+u(e)−∑e∈|c|−l(e).Given any chain d ∈ ZE , there is an associated homomorphism d̂ : C → Z defined by d̂(c) =∑e∈|c| d(e)c(e). Thus d̂ = 0if and only if d ∈ C∗ and d̂1 = d̂2 if and only if d1 − d2 ∈ C∗.∗E-mail address: daniel.slilaty@wright.edu†Seymour proved a special case of Camion’s result in [9, (14.5)] which he cites as a previously known folklore result.‡Seymour famously proved [8] that regular chain groups decompose along small separations into cycle spaces of graphs, cocycle spaces of graphs, andcopies of the chain group associated with the matroid R10.D. Slilaty / Discrete Math. Lett. 12 (2023) 62–65 63Camion’s result may be stated as follows.Theorem 1.1 (Camion [2]). Let C ≤ ZE be a regular chain group, let f : C → Z be a homomorphism, and let I be anassignment Z-intervals to E. The following are equivalent.(1) f(c) ∈ I(c) for each primitive chain c ∈ C.(2) f(c) ∈ I(c) for each chain c ∈ C.(3) There is d ∈ ZE satisfying d̂ = f and d(e) ∈ I(e) for each e ∈ |d|.Define the length of c ∈ ZE as length(c) =∑e∈|c| |c(e)|. This, of course, generalizes the notion of the length of a cyclein a graph. Theorem 1.2 is a “parity-respecting” version of Theorem 1.1. When C is the integer cycle space of a graphand all intervals are [−k, k] for some odd k, Theorem 1.2 specializes to a result of Slilaty [10]. For k = 1, Slilaty’s resultspecializes to a result of Pretzel and Youngs [7]. A notable consequence of this theorem is the chain d obtained in Part (3)is everywhere non-zero.Theorem 1.2. Let C ≤ ZE be a regular chain group, let f : C → Z be a homomorphism for which f(c) ≡ length(c) (mod 2)for each c ∈ C, and let I be an assignment Z-intervals to E for which both u(e) and l(e) are odd for each e ∈ E. The followingare equivalent.(1) f(c) ∈ I(c) for each primitive chain c ∈ C.(2) f(c) ∈ I(c) for each chain c ∈ C.(3) There is d ∈ ZE satisfying d̂ = f and d(e) ∈ I(e) with d(e) odd for each e ∈ |d|.Theorem 1.2 will be proven in Section 2. In relation to the discussion in this paper, it is relevant to note that Ghouila-Houry’s Theorem (or Theorem 1.1) can be used to give a new proof in the non-trivial direction of the elegant Minty-VitaverTheorem [6,12] stated below. We will give this proof in Section 3. An acyclic orientation f of a graph G partitions the edgesof any cycle C in G into two nonempty sets, C+ and C−. Letratio(C, f) = max{|C+||C−|,|C−||C+|}.Theorem 1.3 (Minty [6] and Vitaver [12].). A graph G is k-colorable if and only if there is an acyclic orientation f of G forwhich ratio(C, f) ≤ k− 1 for all cycles C in G.2. Proof of Theorem 1.2Chain c′ conforms to chain c when |c′| ⊆ |c| and c′(e)c(e) ≥ 0 for all e ∈ E. Tutte proved [11, (5.43)] that every c in a regularchain group C has a decomposition c = c1 + · · · + cm in which each ci ∈ C is primitive and conforms to c. Our proof ofTheorem 1.2 is much like Seymour’s proof in [9, (14.5)] with the added concern of keeping track of parities.Proof of Theorem 1.2. (1→ 2) Any c ∈ C decomposes as c = c1+ · · ·+cm in which each ci is primitive and conforms to c. Nowbecause each ci conforms to c and satisfies f(ci) ∈ I(ci) we may add all of the associated inequalities to obtain f(c) ∈ I(c).(2→ 1) Follows trivially from the fact that primitive chains are chains.(3→ 1) Suppose that d ∈ ZE satisfies (3). Then, for any primitive c ∈ Cf(c) =∑e∈Ec(e)d(e) =∑e∈|c|+d(e)−∑e∈|c|−d(e) ≤ u(c).Similarly, f(c) ≥ l(c) and so f(c) ∈ I(c), as required.(1 → 3) Suppose that f(c) ∈ I(c) for all c ∈ C. We proceed by induction on the quantity L(I) =∑e∈E(u(e) − l(e)). IfL(I) = 0, then u(e) = l(e) for all e and so f(c) ∈ I(c) implies that I(c) = [f(c), f(c)] for each c. Thus∑e∈Eu(e)c(e) = f(c),which along with the fact that u(e) = l(e) is odd for each e ∈ E makes u = l our desired element of ZE .63D. Slilaty / Discrete Math. Lett. 12 (2023) 62–65 64Now take interval assignment I such that L(I) > 0. Thus there is e0 ∈ E for which u(e0) − l(e0) > 0. Since l(e0) andu(e0) are both odd, u(e0)− l(e0) ≥ 2. Let I ′ be an interval assignment such that I ′(e) = I(e) for all e 6= e0 andI ′(e0) = [l(e0), u(e0)− 2].Also, let I ′′ be an interval assignment such that I ′′(e) = I(e) for all e 6= e0 and I ′′(e0) = [l(e0) + 2, u(e0)]. EvidentlyL(I ′) = L(I ′′) = L(I)− 2 and for all e ∈ E the intervals I ′(e) and I ′′(e) have odd-valued endpoints. So, if either f(c) ∈ I ′(c)for all primitive c ∈ C or f(c) ∈ I ′′(c) for all primitive c ∈ C, then by induction we get d ∈ ZE satisfying (3). If neither of theseconditions hold, then there are primitive chains c1, c2 ∈ C with e0 ∈ |c1|+ ∩ |c2|+ such that f(c1) /∈ I ′(c1) and f(c2) /∈ I ′′(c2).Because f(c) ≡ length(c) (mod 2) for all c ∈ C and l(e) and u(e) are odd for each e ∈ E,f(c) ≡ length(c) ≡ l(c) ≡ u(c) (mod 2)for all c ∈ C. So now because f(c1) ∈ I(c1) and f(c2) ∈ I(c2) we get thatf(c1) = u(c1) =∑e∈|c1|+u(e)−∑e∈|c1|−l(e) andf(c2) = l(c2) =∑e∈|c2|+l(e)−∑e∈|c2|−u(e).Replace c2 with −c2 to obtainf(c2) = u(c2) =∑e∈|c2|+u(e)−∑e∈|c2|−l(e)where we now have e0 ∈ |c1|+ ∩ |c2|−. Now let z = c1 + c2. Since we have already proven 1 ←→ 2 we get the following inwhich A = (|c1|+ ∩ |c2|−) ∪ (|c1|− ∩ |c2|+):f(z) ≤ u(z)=∑e∈|z|+z(e)u(e) +∑e∈|z|−z(e)l(e)=∑e∈|c1|+u(e)−∑e∈|c1|−l(e) +∑e∈|c2|+u(e)−∑e∈|c2|−l(e)−∑e∈A(u(e)− l(e))= f(c1) + f(c2)−∑e∈A(u(e)− l(e))= f(z)−∑e∈A(u(e)− l(e)).Now because e0 ∈ A we obtain 0 ≤ −∑e∈A(u(e)− l(e)) ≤ −2, a contradiction.3. A new proof of the Minty-Vitaver TheoremSay that f is an acyclic orientation of G for which ratio(C, f) ≤ k− 1 for all cycles C. Let C be the integer cycle space of thedirected graph (G, f) and assign to each e the interval [1, k − 1]. Now and let c ∈ C be any primitive and also elementarychain. (We will consider all primitive chains later.) Thus |c| corresponds to the edge set of a cycle, call it C, in G wherewithout loss of generality |c|+ = C+ and |c|− = C− with respect to f . By assumption we have that |C+|/|C−| and |C−|/|C+|are both at most k − 1. Thus we have thatl(c) =∑e∈|C+|1−∑e∈|C−|(k − 1) = |C+| − (k − 1)|C−| ≤ |C+| − |C+| = 0and also thatu(c) =∑e∈|C+|(k − 1)−∑e∈|C−|1 = (k − 1)|C+| − |C−| ≥ |C−| − |C−| = 0.Thus l(c) ≤ 0 ≤ u(c) for any primitive and elementary c ∈ C. Now if c is any primitive chain in C, then because c isorthogonal to all of the chains in the cocycle space C∗ (in particular, the vertex cocycles) |c| corresponds to a subgraph ofthe directed graph (G, f) in which each vertex has equal in-degree and out-degree. Thus |c| partitions into edge-disjointdirected cycles in (G, f), thus c = c1 + · · · + cn in which each ci is primitive, elementary, and conforms to c. So now since64D. Slilaty / Discrete Math. Lett. 12 (2023) 62–65 65we have proven that l(ci) ≤ 0 ≤ u(ci) for each ci, we now have that l(c) ≤ 0 ≤ u(c). Thus Ghouila-Houry’s Theorem (whichis a special case of Theorem 1.1 within the ring of integers) implies that there is d ∈ C∗ for which d(e) ∈ [1, k − 1] for eachedge e in G. A chain in C∗ is often called a tension in graph theory. Thus d (mod k) is a nowhere-zero Zk-tension. It is wellknown that, for a connected graph G, the proper k-colorings of G using color set Zk correspond k-to-1 to the nowhere-zeroZk-tensions of G. (See, for example, the discussion of potential functions in [1, Sec. 5.3].)References[1] C. Berge, Graphs and Hypergraphs, Revised Edition, North-Holland, Amsterdam, 1976.[2] P. Camion, Matrices Totalement Unimodulaires et Problmes Combinatoires, Ph.D. Thesis, Université Libre de Bruxelles, Brussels, 1963.[3] P. Camion, Unimodular modules, Discrete Math. 306 (2006) 2355–2382.[4] A. Ghouila-Houri, Sur l’existence d’un flot ou d’une tension prenant ses valeurs dans un groupe abélien, C. R. Acad. Sci. Paris 250 (1960) 3931–3933.[5] A. J. Hoffman, Some recent applications of the theory of linear inequalities to extremal combinatorial analysis, In: R. Bellman, M. Hall Jr. (Eds.), Combinatorial Analysis,Proc. Sympos. Appl. Math. 10, American Mathematical Society, Providence, 1960, 113–127.[6] G. J. Minty, A theorem on n-coloring the points of a linear graph, Amer. Math. Monthly 69 (1962) 623–624.[7] O. Pretzel, D. Youngs, Cycle lengths and graph orientations, SIAM J. Discrete Math. 3 (1990) 544–553.[8] P. D. Seymour, Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980) 305–359.[9] P. D. Seymour, Matroids and multicommodity flows, European J. Combin. 2 (1981) 257–290.[10] D. Slilaty, Integer functions on the cycle space and edges of a graph, Graphs Combin. 26 (2010) 293–299.[11] W. T. Tutte, Lectures on matroids, J. Res. Nat. Bur. Stand. Sect. B 69 (1965) 1–47.[12] L. M. Vitaver, Determination of minimal coloring of vertices of a graph by means of Boolean powers of the incidence matrix, Dokl. Akad. Nauk SSSR 147 (1962) 758–759.65

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