Mixed Cages: monotony, connectivity and upper bounds

(2) Universidad Aut\'onoma Metropolitana; (3) Departamento de Matem\'aticas Aplicadas y Sistemas; Claudia de la Cruz (2); Diego Gonz\'alez-Moreno (3) ((1) Instituto de Matem\'aticas; Gabriela Araujo-Pardo (1); Universidad Aut\'onoma Metropolitana-Cuajimalpa); Universidad Nacional Aut\'onoma de M\'exico

REVIEW 2 major objections 5 minor 16 references

Mixed Cages: monotony, connectivity and upper bounds

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Pith's one-line read Mixed cages grow strictly larger as girth increases, for arc-degree 1 or 2.

desk verdict Solid structural results for mixed cages, but the monotonicity proof has a repairable gap and the lower bound section is not proven as written. read the letter →

arxiv 2009.13709 v1 pith:OE4H53G3 submitted 2020-09-29 math.CO

Gabriela Araujo-Pardo (1) , Claudia de la Cruz (2) , Diego González-Moreno (3) ((1) Instituto de Matemáticas , Universidad Nacional Autónoma de México , (2) Universidad Autónoma Metropolitana , (3) Departamento de Matemáticas Aplicadas y Sistemas , Universidad Autónoma Metropolitana-Cuajimalpa) This is my paper · ORCID

classification math.CO MSC 05C3505C4005C20

keywords mixedcagesmonotonicityconnectivitybiaffineplanescirculantdigraphsgirthregulargraphsdirected

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies mixed graphs that are z-regular in directed arcs, r-regular in undirected edges, and have girth g, and asks how small such a graph can be. It proves that for z = 1 or 2 this minimum order is strictly increasing in g, so a larger prescribed girth genuinely forces more vertices. The same monotonicity is then used to show that the underlying undirected graph of every such mixed cage is 2-connected, and a separate counting argument shows that every mixed cage is strongly connected. The paper also provides lower and upper bounds and settles one exact value: a [10,3;5]-mixed cage has order 50. These are basic structural facts for mixed cages, the directed-and-undirected analogue of classical cages, and they constrain any future search for smallest examples.

What carries the argument

The load-bearing mechanism is the local structure of a shortest cycle: Lemma 1 asserts that in any [z,r;g]-mixed cage with z ∈ {1,2}, some girth cycle has either two consecutive arcs or two consecutive edges. Starting from that pattern, the proofs of Theorem 3 delete one or two vertices and add a carefully chosen set of edges and arcs, preserving the prescribed arc-degree z and edge-degree r while forcing the girth down exactly to g. For connectivity, strict monotonicity does the heavy lifting: the splicing operation that forms a smaller mixed graph from a cut component and its reverse copy is only a contradiction because n[z,r;·] is strictly increasing. For the upper bounds, the constructions use the incidence graph of a biaffine plane over a prime field, reduced to a smaller bipartite graph and amalgamated with a circulant digraph that contributes the z arcs per vertex; a second construction lays g copies of an (r,g)-cage in a cycle and joins consecutive copies by directed 1-factors.

What would settle it

Search all [1,r;g]- and [2,r;g]-mixed cages for small r and g and check whether any has every shortest cycle strictly alternating between arcs and edges; one such example would falsify Lemma 1 and the proof of Theorem 3. For the exact order claim, a computer search for a [10,3;5]-mixed graph on at most 49 vertices, or an independent verification that the 50-vertex graph displayed in the paper has girth exactly 5, would settle Corollary 12.

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Extended reading notes

Core claim

The central claim is that the order function n[z,r;g] of a mixed cage is strictly monotone in g for z ∈ {1,2}: whenever 3 ≤ g1 < g2, one has n[z,r;g1] < n[z,r;g2]. The proof passes from a [z,r;g+1]-cage to a smaller [z,r;g]-graph by deleting one vertex (or two vertices adjacent on the chosen cycle) and reconnecting the incident edges and arcs in pairs; for the operation to start, the cage must contain a shortest cycle with two consecutive arcs or two consecutive edges, which Lemma 1 guarantees for z ∈ {1,2}. Monotonicity is then the engine for Theorem 5: if some vertex were a cut vertex of the underlying graph, taking a smallest component and gluing it to a reversed copy would produce a mixed graph with fewer vertices and girth at least g, contradicting the strict inequality n[z,r;g'] < n[z,r;g]. The paper also establishes strong connectivity of every mixed cage by counting arcs across strong components.

Load-bearing premise

The proof of strict monotonicity rests on Lemma 1's guarantee that every [z,r;g]-mixed cage with z ∈ {1,2} contains a shortest cycle with two consecutive arcs or two consecutive edges; if a cage with all shortest cycles strictly alternating between arcs and edges existed, the vertex-deletion operations would have no guaranteed starting pattern and the monotonicity proof would not go through.

Editorial extensions

If this is right

For z ∈ {1,2}, n[z,r;g] is strictly increasing in g, so two mixed cages with the same r but different girths cannot share an order.
The underlying graph of every [z,r;g]-mixed cage with z ∈ {1,2} is 2-connected; cutting any single vertex leaves it connected.
Every [z,r;g]-mixed cage is strongly connected in the directed sense, regardless of the degree parameters.
There is a [10,3;5]-mixed cage with exactly 50 vertices, matching the lower and upper bounds proved in the paper.
For z ≠ 2 and r in the stated range, there are [z,r;5]-mixed graphs of order 2pr, where p is the smallest prime with 4z+1 ≤ p ≤ 5z; and for every z' ≤ n0(r,g) there are [z',r;g]-mixed graphs of order g n0(r,g).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The monotonicity proof leans entirely on Lemma 1's 'two consecutive arcs or two consecutive edges' pattern; whether the same pattern holds for z ≥ 3 is the natural next test, and a counterexample there would stop the proof method, not necessarily the theorem.
The biaffine-plane construction is run over prime fields to guarantee the needed prime in [4z+1, 5z], but the plane itself exists over any finite field; trying composite q could produce [z,r;5] graphs of order 2qr for parameter pairs the current statement misses.
The exact value 50 for [10,3;5] shows the general lower bound n0(r,g)+2z is far from tight in at least one case; identifying parameters where that bound is sharp would give a cleaner picture of when the easy counting argument suffices.
If strict monotonicity could be proved for z ≥ 3 by other means, the splicing argument would immediately promote the 2-connectivity theorem to all arc-degrees, since the rest of Theorem 5 does not use z.

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Formalized claims in Lean

Claim #1: The central claim is that the order function n[z,r;g] of a mixed cage is strictly monotone in g for z ∈ {1,2}: whenever 3 ≤ g1 < g2, one has n[z,r;g1] < n[z,r;g2]. The proof passes from a [z,r;g+1]-cage to a smaller [z,r;g]-graph by deleting one vertex (or two vertices adjacent on the chosen cycle) and reconnecting the incident edges and arcs in pairs; for the operation to start, the cage must con

/-- @claim 1 The central claim is that the order function n[z,r;g] of a mixed cage is strictly monotone in g for z ∈ {1,2}: whenever 3 ≤ g1 < g2, one has n[z,r;g1] < n[z,r;g2]. The proof passes from a [z,r;g+1]-cage to a smaller [z,r;g]-graph by deleting one vertex (or two vertices adjacent on the chosen cycle) and reconnecting the incident edges and arcs in pairs; for the operation to start, the cage must con -/
def central_claim : Prop :=

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First note: this is a real step forward for mixed cages. The monotonicity theorem for z=1,2, the strong-connectivity result, and the 2-connectivity theorem are the first structural results of their kind, and the 50-vertex [10,3;5] example is a concrete data point. The authors are working in an established program, and the constructions adapt known techniques in a sensible way.

The proofs are not machine-checked and have a few under-derived steps. The most important is Lemma 1, the base of Theorem 3. In Case 1, the inequality d_{G-y}(alpha_i,alpha_j)+1 >= g is asserted, but girth only gives d+2 >= g, hence d >= g-2. The gap is real. I think it is repairable: if d = g-2, the path plus the two incidences through y is a girth cycle with two consecutive edges at y, contradicting the alternating assumption. The same repair applies to the 'similar analysis' cases. But as written, the monotonicity proof has a hole.

Bigger issue is Proposition 6. The sentence 'since every vertex has z ex-neighbors and z in-neighbors, it follows that |V(G)| >= n0(r,g)+2z' is not a valid inference. The ex-neighbors and in-neighbors are vertices already counted in the underlying graph; adding arcs does not force new vertices. The Moore bound n0(r,g) counts vertices in the r-regular graph; the arc-degree does not enter. This lower bound looks unproven and I suspect it may be false in general. Theorem 7 uses Proposition 6 to get the initial 30-vertex bound, so the final 50 lower bound is not established as written. The final counting step in Theorem 7 also needs a proof that the two 10-sets are disjoint; the text asserts it without argument.

Minor points: Theorem 5's inequality should be 2(g-2)+2 > g, not 2(g-2)>g. The prime-check in Theorem 10 for z<=3275 is hand-waved and should be documented. Theorem 7's case analysis is long and partially delegated to figures and 'similarly', so it needs careful checking.

Overall: the structural theorems (strong connectivity, 2-connectivity) are sound and likely correct. The monotonicity theorem is probably correct and repairable. The lower bound section is not in a citable state. This paper deserves a serious referee and a round of major revision, not a desk rejection. The upper-bound constructions and the 50-vertex example are worth preserving even if the lower bound is tightened later.

Referee Report

2 major / 5 minor

Summary. The paper studies [z,r;g]-mixed cages, i.e., minimal mixed graphs that are z-regular in arcs and r-regular in edges with girth g. The main results are: (1) a monotonicity theorem n[z,r;g1] < n[z,r;g2] for z in {1,2} and g1 < g2; (2) strong connectivity of every mixed cage; (3) 2-connectivity of the underlying graph for z in {1,2}; (4) a lower bound n[z,r;g] >= n0(r,g)+2z; (5) an upper bound family n[z,r;5] <= 2pr via amalgams of biaffine planes and circulant digraphs; and (6) an upper bound n[z',r;g] <= g n0(r,g). These are combined to exhibit a [10,3;5]-mixed cage of order 50.

Significance. The monotonicity result, if fully established, is a natural mixed-graph analogue of the classical monotonicity of cage orders and is used to derive a 2-connectivity result, so it is the conceptual core of the paper. The lower bound and the two upper-bound constructions are concrete and make the paper self-contained enough to certify the order-50 [10,3;5]-cage. The proof of the lower bound for the [10,3;5] case is a nontrivial case analysis. The paper also gives a constructive family for girth 5 parameterized by a prime near 4z, and the general girth upper bound via directed blow-ups of cages. The authors rely on standard external results (Moore bounds, existing cages, biaffine planes, Dusart's prime gap) and include a finite computational check for small z, which are appropriate. However, the proof of the key Lemma 1 contains an unjustified inequality, and Proposition 6 is under-proved; these need to be repaired before the main claims are fully supported.

major comments (2)

[Section 2.1, Lemma 1] In the proof of Lemma 1, Case 1 (z=1, r even), the assertion ell(C') >= d_{G-y}(alpha_i, alpha_j)+1 >= g is not justified by the girth hypothesis alone. The girth argument gives only d_{G-y}(alpha_i, alpha_j) >= g-2, because the path in G-y together with the two incidences through y is a closed walk of length d+2 in a girth-g graph. If d = g-2, this closed walk is a g-cycle containing two consecutive arcs (or two consecutive edges), which contradicts the lemma's standing assumption that all g-cycles are alternating. This additional contradiction is missing from the proof, and the same gap is implicit in the 'similar analysis' claims for the r-odd and z=2 cases. Since Lemma 1 is the starting point of the proof of Theorem 3 and hence of Theorem 5, this step must be completed.
[Section 3.1, Proposition 6] The proof of Proposition 6 asserts |V(G)| >= n0(r,g)+2z after noting that every vertex has z out-neighbors and z in-neighbors. This does not follow as written: one must prove that the sets N^+(v), N^-(v), and the Moore-bound ball in G-A(G) around v are pairwise disjoint. The disjointness does follow from the girth condition (an in- or out-neighbor inside the ball would create a cycle of length at most floor((g-1)/2)+1 < g, and a vertex that is both an in- and out-neighbor would create a 2-cycle), but this argument is absent. Without it, the '+2z' term is unsupported. Please supply the missing argument or rephrase the proof.

minor comments (5)

[Section 2.2, Theorem 5] The displayed inequality g(G*) >= 2(g-2) > g is false for g=3 and g=4. The intended counting gives 2d+2 with d >= g-2, hence a lower bound of 2g-2, which is greater than g for all g >= 3; please correct the displayed inequality.
[Section 3.2.1, Theorem 10] The statement allows r in {1,...,p}, but the proof treats r in {2,...,p-1} via gamma in {1,...,p-2} and separately r=p. For r=1 one would need gamma=p-1, which is outside the stated range; please either adjust the statement or add the missing argument for r=1.
[Throughout] There are several typos and wording issues: 'the More bound' should be 'the Moore bound' (Section 3.1), 'monotonocity' should be 'monotonicity' (Section 4), and 'ex-neighbors' should likely be 'out-neighbors'.
[Section 2.1, Lemma 1] The phrase 'Since C is an induced cycle' should be justified; in a graph of girth g, any cycle of length g is chordless, but this fact is not stated in the paper.
[Section 3.2.2, Theorem 11] The notation '->Ai(j) = union_{j}^{i=1} Ai' is hard to parse; please define A_i explicitly as the i-th oriented 1-factor and write the union as the first j factors, so that the construction is unambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main estimates are separate from the claims; the only self-citation is a base case from prior published work.

full rationale

The paper's derivation chain is self-contained with respect to the claims it labels as predictions. Lemma 1 and Theorem 3 use standard contradiction constructions: from a putative minimal [z,r;g]-cage, a smaller [z,r;g]-graph is produced by vertex deletion plus edge/arc replacement, and the contradiction is to cage minimality, not to the monotonicity being proved. The lower and upper bounds (Propositions 6 and 7, Theorems 10 and 11) are derived from external ingredients — Moore bounds, existence of undirected cages, biaffine plane incidence graphs, circulant digraphs, and Dusart's prime-gap estimate — none of which are defined in terms of n[z,r;g] or presuppose monotonicity. The use of prior paper [5] for the base case n[1,r;3]=r+3 and n[1,r;4]≥2r+4 is a citation of a published lower-bound result; it is not an input renamed as the conclusion, and it does not carry the central argument, which is established for g≥4 by the deletion/replacement construction. Theorem 5 depends on Theorem 3 as a premise, but that is ordinary theorem dependency, not circularity, because Theorem 3 is proved separately. The only flagged issue in review is a possible off-by-one in a distance claim inside Lemma 1; even if correct, that would be a proof-correctness gap, not a circular reduction. No fitted parameter is called a prediction, and no uniqueness theorem is imported from the authors' own prior work to forbid alternatives. Therefore the paper merits a circularity score of 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard cage theory (Moore bounds, existence of undirected cages), on the biaffine-plane incidence graph, and on a prime-gap fact. The prime-gap fact is the only computational assumption whose verification is not shown in the paper. No free parameters or invented entities appear.

assumptions (4)

standard math Moore bound for r-regular graphs of girth g
Used in Proposition 6 and Theorem 11 as an external lower bound on the order of the edge-only subgraph; standard in cage theory.
standard math Existence of (r,g)-cages
Theorem 11 builds g copies of an (r,g)-cage H; existence is cited to Erdos and Sachs [10].
standard math Incidence graph properties of the biaffine plane
Remark 8, attributed to Brown [8], gives the q-regular bipartite graph Bq of order 2q^2 with girth 6 or 8, used in Theorem 10.
domain assumption For every positive integer z != 2 there is a prime p with 4z+1 <= p <= 5z
Theorem 10 depends on this prime-gap fact. It is proven for z >= 3275 via Dusart [9] and asserted for 1 <= z <= 3275 via unspecified computer calculations.

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Pith. "Pith review of Mixed Cages: monotony, connectivity and upper bounds." pith.science (2026). https://pith.science/paper/OE4H53G3

@misc{pith2026200913709,
  author       = {Pith},
  title        = {Pith review of: Mixed Cages: monotony, connectivity and upper bounds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OE4H53G3}},
  note         = {Machine review of arXiv:2009.13709}
}

abstract

A \emph{$[z, r; g]$-mixed cage} is a mixed graph $z$-regular by arcs, $r$-regular by edges, with girth $g$ and minimum order. %In this paper we study structural properties of mixed cages: Let $n[z,r;g]$ denote the order of a $[z,r;g]$-mixed cage. In this paper we prove that $n[z,r;g]$ is a monotonicity function, with respect of $g$, for $z\in \{1,2\}$, and we use it to prove that the underlying graph of a $[z,r;g]$-mixed cage is 2-connected, for $z\in \{1,2\}$. We also prove that $[z,r;g]$-mixed cages are strong connected. We present bounds of $n[z,r;g]$ and constructions of $[z,r;5]$-mixed graphs and show a $[10,3;5]$-mixed cage of order $50$.

Figures

Figures reproduced from arXiv: 2009.13709 by the authors.

**Figure 1.** Construction of G0 from a [1, r; g]-mixed cage with r even. we define a set A0 depending on the the sets N −(u 0 ) and N −(s 0 ). If u ∈ N −(u 0 ) or s ∈ N −(s 0 ), then A0 = { −→ us0 , −→ su0}. In other case, A0 = { −→ uu0 , −→ ss0}. Define G0 as G0 = G − y + E0 + A0 (see [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Construction of G0 from a [2, r; g]-mixed cage with r even and −→ su0 ∈ A(G). If r is odd, let w = vr, N(w) = {v 0 1 , . . . , v0 r}, where v 0 r = y, N −(w) = {x 0 , x00} and N +(w) = {y 0 , y00}. Let E0 = {v2i−1v2i , v0 2i−1 v 0 2i : 1 ≤ i ≤ (r − 1)/2}. Since d −(u 0 ) = d −(s 0 ) = 2, it follows that |N −(u 0 )∩ {u, s}| ≤ 1 and |N −(s 0 ) ∩ {u, s}| ≤ 1. Define a set Ay depending on the sets N −(u 0 ) and N −(s 0 … view at source ↗

**Figure 3.** Operation in a [1, r; g + 1]-mixed cage with r even, in a cycle with two edges consecutive. Let G0 = G − u + E0 . Observe that g(G0 ) ≤ g, since G0 contains the cycle C − u + v1v2. We claim that g(G0 ) = g. Let C 0 be a cycle of G0 such that `(C 0 ) = g(G0 ) ≤ g. If E(C 0 ) ∩ E0 = ∅, then E∗ (C 0 ) ⊆ E(G), implying that `(C 0 ) ≥ g + 1, a contradiction. Hence, E∗ (C 0 ) ∩ E0 6= ∅. If |E∗ (C 0 ) ∩ E0 | = 1, then E∗ (… view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: Operation in a [1, r; g + 1]-mixed cage with r odd, in a cycle with two arcs consecutive. The case in which C contains two consecutive arcs is analogous. Case 2) Suppose z = 2. Let N −(u) = {x1, x2} and N +(u) = {y1, y2}. Case 2.1) Suppose that r is even. If v1u, uv2 ∈…

**Figure 5.** Figure 5: Operation in a [2, r; g + 1]-mixed cage with r even, in a cycle with two edges consecutive and there is no the arc −−→x1y1 or −−→x2y2. Suppose now that −−→x1u, −→uy1 ∈ A(C) and y2 ∈ N +(x2). If y 0 ∈ N +(y2) ∩ V (C), let E0 = {v2i−1v2i : 1 ≤ i ≤ r/2} ∪ {−−→x1y2, −−→x2y…

**Figure 6.** Figure 6: Operation in a [2, r; g + 1]-mixed cage with r even, in a cycle with two arcs consecutive and there is no the arc −−→x2y2. that |N −(yi) ∩ {x1, x2}| ≤ 1. If either x1 ∈ N −(y1) or x2 ∈ N −(y2), then Au = { −−→x1y2, −−→x2y1}. In other case set Au = { −−→x1y1, −−→x2y2}. …

**Figure 7.** Figure 7: Operation in a [2, r; g + 1]-mixed cage with r odd, in a cycle with two consecutive arcs and there is the arc −−→x2y2. that [V (H1), V (H∗ )] 6= ∅. The number of arcs of G is |A(G)| = |V (G)|z = (|V (H1)| + |V (H∗ )|)z. On the other hand, for every vertex v ∈ V (H1) it…

**Figure 8.** Figure 8: Structure with two adjacent edges. Notice that if we want maximize the number of edges and the arcs in G[Z] (the subgraph induced graph by Z), there are only nine possibilities for G[Z] (see [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: The nine configurations of G[Z]. Similarly, in the possibilities of G[Z] that have a vertex w with | −→N (w) ∩ Z| = 0, it follows that | −→N (w) ∩ (N +(u) \ {z})| ≥ 3. Since x1, x2, y /∈ −→N (w), a contradiction is obtained. Consequently, there are no two incident edge…

**Figure 10.** Figure 10: Structure with at least 11 vertices in N +(u). x, y /∈ ( −→N (z) ∩ N +(u)). Let Z = −→N (z) ∩ N +(u). In this case we only have five possibilities for G[Z] (see [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]

**Figure 11.** Figure 11: The five configurations of four vertices with the maximum number of edges and arcs preserving [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

**Figure 12.** Figure 12: A [3, 13; 5]-mixed graph. be a 1-factor (oriented) of B(Hi , Hi+1). Let −→Ai(j) = Sj i=1 Ai and let G∗ = G + Sg−1 i=0 −→Ai(z 0 ). Observe that we can always get that Ai ∩ Aj = ∅, it follows that G∗ is a [z 0 , r; g]-mixed graph. In the following we will prove that G∗ …

**Figure 13.** Figure 13: A [10, 3; 5]-mixed cage of order 50. 4 Future work The problem of find a mixed cage and study their properties is very recently. As a suggestion to continue with the topic we propose two problems: 12 [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

Discrete Math

Abajo E., Araujo-Pardo G., Balbuena C., Bendala M., New small regular graphs of girth ﬁve. Discrete Math. 340, no. 8, 18781888 (2017)

work page 2017
[2]

In preparation

Abajo E., Araujo-Pardo G., A note on new small mixed graphs of girth ﬁve. In preparation

work page
[3]

312 (2012) 2832-2842

Abreu M., Araujo-Pardo G., Balbuena C., Labbate D., Families of small regular graphs of girth 5, Discrete Math. 312 (2012) 2832-2842

work page 2012
[4]

Discrete Applied Math

Araujo-Pardo G., Balbuena C., Miller, M., ˇZd´ ımalov´ a M.: A family of mixed graphs with large order and diameter 2. Discrete Applied Math. 218, 57–63 (2017)

work page 2017
[5]

Graphs and Combi- natorics

Araujo-Pardo G., Hern´ andez-Cruz C., Montellano-Ballesteros J.J., Mixed Cages. Graphs and Combi- natorics. 35 No. 5, 989-999 (2019). https://doi.org/10.1007/s00373-019-02050-1

work page doi:10.1007/s00373-019-02050-1 2019
[6]

Balbuena C., Salas J., A new bound for the connectivity of cages, Applied Mathematics Letters, 25(11) (2012) 16761680 (2012)

work page 2012
[7]

308(15), 32493256 (2008)

Balbuena C., Lin Y., Marcote X., Miller M., On the connectivity of (k, g)-cages of even girth, Discrete Math. 308(15), 32493256 (2008)

work page 2008
[8]

Brown W.G., On Hamiltonian regular graphs of girth six, J. Lond. Math. Soc. 42 (1967) 514520

work page 1967

Show all 16 references

[9]

Thesis, Universit´ e de Limoges, 1998

Dusart P., Autour de la fonction qui compte le nombre de nombres premiers, Ph.D. Thesis, Universit´ e de Limoges, 1998

1998
[10]

and Sachs

Erd¨ os P. and Sachs. H., Regul¨ are Graphen gegebener Taillenweite mit minimaler Knotenzahl. Wiss. Z. Uni. Halle (Math. Nat.), 12, 251–257, (1963)

1963
[11]

Exoo, G., Jaycay, R., Dynamic cage survey, Electronic Journal of Combinatorics, 15, #DS16, (2008)

2008
[12]

and Rodger C.A., Connectivity of Cages, J

Fu H.L., Huang K.C. and Rodger C.A., Connectivity of Cages, J. Graph Theory 24, 187-191, 1997

1997
[13]

Journal of Graph Theory

Lin Y., Miller M., Rodger C., All ( k;g)cages are kedgeconnected. Journal of Graph Theory. 48. 219 -
[14]

W.T., A family of cubical graphs

Tutte. W.T., A family of cubical graphs. Math. Proc. Cambridge Philos. Soc., 43 (4), 459-474, (1947)

1947
[15]

Electron J Combin

Wang P., Baoguang X., Wang J., A Note on the Edge-Connectivity of Cages. Electron J Combin. 10. 10.37236/1742. (2003). 13

2003 doi
[227]

10.1002/jgt.20052, (2005)

2005 doi

Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [1]

Discrete Math

Abajo E., Araujo-Pardo G., Balbuena C., Bendala M., New small regular graphs of girth ﬁve. Discrete Math. 340, no. 8, 18781888 (2017)

work page 2017

[2] [2]

In preparation

Abajo E., Araujo-Pardo G., A note on new small mixed graphs of girth ﬁve. In preparation

work page

[3] [3]

312 (2012) 2832-2842

Abreu M., Araujo-Pardo G., Balbuena C., Labbate D., Families of small regular graphs of girth 5, Discrete Math. 312 (2012) 2832-2842

work page 2012

[4] [4]

Discrete Applied Math

Araujo-Pardo G., Balbuena C., Miller, M., ˇZd´ ımalov´ a M.: A family of mixed graphs with large order and diameter 2. Discrete Applied Math. 218, 57–63 (2017)

work page 2017

[5] [5]

Graphs and Combi- natorics

Araujo-Pardo G., Hern´ andez-Cruz C., Montellano-Ballesteros J.J., Mixed Cages. Graphs and Combi- natorics. 35 No. 5, 989-999 (2019). https://doi.org/10.1007/s00373-019-02050-1

work page doi:10.1007/s00373-019-02050-1 2019

[6] [6]

Balbuena C., Salas J., A new bound for the connectivity of cages, Applied Mathematics Letters, 25(11) (2012) 16761680 (2012)

work page 2012

[7] [7]

308(15), 32493256 (2008)

Balbuena C., Lin Y., Marcote X., Miller M., On the connectivity of (k, g)-cages of even girth, Discrete Math. 308(15), 32493256 (2008)

work page 2008

[8] [8]

Brown W.G., On Hamiltonian regular graphs of girth six, J. Lond. Math. Soc. 42 (1967) 514520

work page 1967