Tight factorizations of girth-g-regular graphs Italo J. Dejter University of Puerto Rico Rio Piedras, PR 00936-8377 [email protected] Abstract The determination of 1-factorizations F of girth-regular graphs of girth, regular degree and chromatic index g is proposed for the cases in which each girth cycle intersects every 1-factor of F . This endeavor may apply to priority assignment problems, managerial situations in optimization and decision making. Applications to hamiltonian decom- posability (via union of pairs of 1-factors) and to 3-dimensional geometry (M¨obius-strip compounds and hollow-triangle polylinks) are given. 1 Introduction Let 3 ≤ g ∈ Z. We inquire whether assigning g colors properly to the edges of a finite graph Γ (i.e. not two adjacent edges of Γ with same color) can be performed so that each girth g-cycle of Γ has a bijection from its edges to their colors. Such an inquiry would be applicable to managerial situations in which a number of agents must participate in committees around round tables, with g stools about each table. A roster of g tasks is handed to each agent at each round table, and such agent would be assigned each of the g tasks but at different tables. This can be interpreted as a two-way (vertex-incidence versus girth-cycle-membership) priority assignment problem of sorting the g colors according to some prioritization hierarchy. This way, an assignment problem is conceived, with potential applications in optimization and decision making. We pass to formalize these ideas. Let Γ be a finite connected g-regular simple graph with chromatic index χ′(Γ) = g. Let g = g(Γ) be the girth of Γ. We say Γ is a g-tight graph if g(Γ) = g = χ′(Γ). In each g-tight graph Γ, it makes sense to look for a proper edge-coloring via g colors, each girth cycle colored via a bijection between girth-cycle edges and the g colors they are assigned. We will arXiv:2102.06956v3 [math.CO] 29 Aug 2021 say that such a coloring is an edge-girth coloring of Γ and, in such a case, that Γ is edge-girth chromatic, or egc for short. We focus on g-tight graphs that are girth-regular, a concept whose definition in [22] we recall now. Let {e1,...,eg} be the set of edges incident to a vertex v. Let(ei) be the number of girth cycles containing an edge ei, for 1 ≤ i ≤ g. Assume (e1) ≤ (e2) ≤ ... ≤ (eg). Let the signature of v be the g-tuple ((e1), (e2),..., (eg)). A graph Γ is said to be girth-regular if all its vertices have a common signature. In such a case, the signature of any vertex of Γ is said to be the signature of Γ. In this work, girth-regular graphs that are g-tight will be said to be girth-g-regular graphs and ((e1), (e2),..., (eg))-graphs, or (e1)(e2) ··· (eg)-graphs, if no confusion arises. In this 3 notation, the prefixes (e1)(e2) ··· (eg) will be (and further denoted) as follows: 222 = 2 and 110 = 120 in Section 2; 3333 = 34, 3322 = 3222 and 2222 = 24 in Section 3; 4443 = 433, 1 3221 = 3221 and 3111 = 313 in Section 4; 1111 = 14 in Section 5 ; 55500 = 5302 and 55555 = 55 in Section 6; 111111 = 16 in Section 7. Extending this context, girth-regular graphs Γ of regular degree g and girth larger than g will be said to be 0g-graphs, or improper (e1)(e2) ··· (eg)-graphs. Edge-girth colorings of (e1)(e2) ... (eg)-graphs are equivalent to 1-factorizations [27] such that the cardinality of the intersection of each 1-factor with each girth cycle is 1. These will be called tight factorizations. The resulting colored girth cycles will be said to be tight or tightly colored. If Γ has a tight factorization, then Γ is egc. Unions of 1-factors of such graphs are investigated In Section 8 for their hamiltonian decomposability. Applications to M¨obius-strip compounds and hollow-triangle polylinks can be found in Section 9. 2 Egc girth-3-regular graphs 3 Theorem 1. [22] There is only one (e1)(e2)(e3)-graph Γ with (e1)(e2)(e3)=222=2 . More- 2 over, this Γ is egc. All other proper (e1)(e2)(e3)-graphs are 1 0-graphs, but not necessarily egc. Proof. This is as the proof of item (1) of Theorem 5.1 in [22]. In order to determine which 120-graphs are egc, let Γ′ =(V ′, E′,φ′) be a finite undirected cubic multigraph. Let e ∈ E′ with φ′(e) = {u, v} and u, v ∈ V ′. Then, e determines two arcs (that is, ordered pairs of end-vertices of e) denoted (e; u, v) and (e; v,u). (If the girth g(Γ′) of Γ′ is larger than 2, then Γ′ is a simple graph, a particular case of multigraph). The following definition is an adaptation of a case of the definition of generalized truncation in [14]. Let A′ denote the set of arcs of Γ′. A vertex-neighborhood labeling of Γ′ is a function ρ : A′ →{1, 2, 3} such that for each u ∈ V ′ the restriction of ρ to the set A′(u)= {(e; u, v) ∈ A′ : e ∈ E′; φ′(e) = {u, v}; v ∈ V ′} of arcs leaving u is a bijection. For our purposes, we require ρ(e; u, v)= ρ(e; v,u), ∀e ∈ E′ with φ′(e)= {u, v}, so that each e ∈ E′ is assigned a well-defined color from the color set {1, 2, 3}. This yields a 1-factorization of Γ′ with three ′ ′ ′ ′ 1-factors that we can call E1, E2, E3 for respective color 1, 2, 3, with E being the disjoint ′ ′ ′ union E1 ∪ E2 ∪ E3. For the sake of examples in Fig. 1, to be presented below, let colors 1,2 and 3 be taken as red, blue and green, respectively. Let K3 be the triangle graph with vertex set {v1, v2, v3}. The triangle-replaced graph ′ ′ ′ ∇(Γ ) of Γ with respect to ρ has vertex set {(ei; u, vi): u ∈ V ;1 ≤ i ≤ 3} and edge set ′ {(ei; u, vi)(ej; u, vj)|vivj ∈ E(K3)}∪{u, vρ(e;u,w))(w, vρ(e;w,u))|e ∈ E ; φ(e)= {u,w}}. ′ 2 Note that ∇(Γ ) is a 1 0-graph. We will refer to the edges of the form (ei; u, vi)(ej; u, vj) as ′ ′ ∇-edges, and to the edges (ei; u, vi)(ej; w, vj), u =6 w, as Γ -edges. Observe that a Γ -edge is incident only to ∇-edges and that each vertex of ∇(Γ′) is incident to precisely one Γ′-edge. This yields the following observation. Lemma 2. [14] Let Γ′ be a finite unidirected cubic multigraph of girth g. Then, for any vertex-neighborhood labeling of Γ′, the shortest cycle in the triangle-replaced graph ∇(Γ′) containing a Γ′-edge is of length at least 2g. 2 We say that Γ′ is a generalized snark if its chromatic index χ′(Γ′) is larger than 3. Two examples of generalized snark are: (i) the Petersen graph and (ii) the multigraph obtained by joining two (2k + 1)-cycles (k ≥ 1) via an extra-edge (a bridge between the two (2k +1)- cycles) and adding k parallel edges to each of the two (2k + 1)-cycles so that the resulting ′ ′ multigraph is cubic, see Fig. 1(r) for k = 5. The triangle-replaced graph Γ1 = ∇(Γ ) of a generalized snark Γ′ will also be said to be a generalized snark, as well as the triangle- ′ ′ ′ ′ replaced graph Γi+1 = ∇(Γi) of Γi, for i =1, 2,..., etc. We will say that Γ is snarkless if it is not a generalized snark. Figure 1: Producing (e1)(e2)(e3)-graphs that are egc Vertex-neighborhood labelings of the examples of Γ′ presented below and represented in Fig. 1 are indicated with the elements 1, 2 and 3 of ρ(A′) interpreted respectively as edge colors red, blue and green. The smallest snarkless Γ′’s are: (a) the cubic multigraph ′ ′ Γa of two vertices and three edges as in Fig. 1(a), with ∇(Γa) as the triangular prism P rism(K3) = K2K3, as in Fig. 1(e), where V (K2) = {0, 1} and stands for the graph ′ cartesian product [19]; (b) the cubic multigraph Γb of four vertices resulting as the edge- 3 ′ disjoint union of a 4-cycle and a 2-factor 2K2 as in Fig. 1(b), with ∇(Γb) as in Fig. 1(j). Given a snarkless Γ′, a new snarkless multigraph Γ′′ is obtained from Γ′ by replacing any edge e with end-vertices say u, v, by the submultigraph resulting as the union of a path ′ ′ ′ ′ ′ P4 =(u,u , v , v) and an extra edge with end-vertices u , v . For example, Γ as in Fig. 1(a) has Γ′′ as in Fig. 1(s) and ∇(Γ′′) as in Fig. 1(t). Using this replacement of an edge e by the ′ ′′ said submultigraph, one can transform Γb with the enclosed blue edge e into a Γb as in Fig. ′′ ′′ 1(c), with ∇(Γb ) as in Fig. 1(k); or with the two enclosed red edges into a Γb as in Fig. ′′ 1(d), with ∇(Γb ) as in Fig. 1(l). The replacing submultigraphs are also shown enclosed in Fig. 1(k) and Fig. 1(l). ′ ′ The triangle-replaced graph ∇(Γ ) of any snarkless (e1)(e2)(e3)-graph Γ , either proper 3 2 3 2 or improper, with (e1)(e2)(e3) ∈{2 , 1 0, 0 }, yields an egc 1 0-graph, as illustrated via the ′ ′ four graphs in Fig.