PREPRINT
國立臺灣大學 數學系 預印本 Department of Mathematics, National Taiwan University
www.math.ntu.edu.tw/ ~ mathlib/preprint/2014- 01.pdf
Strong Edge-Coloring for Jellyfish Graphs
Gerard Jennhwa Chang, Sheng-Hua Chen,Chi-Yun Hsu,Chia-Man Hung, and Huei-Ling Lai
February 23, 2014
Strong edge-coloring for jellyfish graphs
Gerard J. Chang
12∗Sheng-Hua Chen
1†Chi-Yun Hsu
1‡Chia-Man Hung
3§Huei-Ling Lai
4¶1Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan
2National Center for Theoretical Sciences, Taipei Office, Taipei, Taiwan
3Ecole Polytechnique, Route de Saclay, 91128 Palaiseau, France´
4Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30010, Taiwan
February 23, 2014
Abstract
A strong edge-coloring of a graph is a function that assigns to each edge a color such that two edges within distance two apart receive different colors. The strong chromatic index of a graph is the minimum number of colors used in a strong edge-coloring. This paper determines strong chromatic indices of cacti, which are graphs whose blocks are cycles or complete graphs of two vertices.
The proof is by means of jellyfish graphs.
1 Introduction
The coloring problem considered in this article has restrictions on edges within distance two apart. The distance between two edges e and e′ in a graph is the minimum k for which there is a sequence e1, e2, . . . , ek of distinct edges such that e = e1, e′ = ek, and ei−1 shares an end vertex with ei for 2 ≤ i ≤ k. A strong
∗E-mail: gjchang@math.ntu.edu.tw. Supported in part by the National Science Council under grant NSC98-2115-M-002-013-MY3.
†E-mail: b91201040@ntu.edu.tw.
‡E-mail: b99201035@ntu.edu.tw.
§E-mail: mandy00724@gmail.com.
¶E-mail: mimimandy135.am98@g2.nctu.edu.tw.
edge-coloring of a graph is a function that assigns to each edge a color such that any two edges within distance two apart receive different colors. A color class of a strong edge-coloring is the set of all edges using the same color. A strong k-edge-coloring is a strong edge-coloring using at most k colors. An induced matching is an edge set in which two distinct edges are of distance at least two. Finding a strong k-edge-coloring is equivalent to partitioning the edge set of the graph into k induced matchings. The strong chromatic index of a graph G, denoted by χ′s(G), is the minimum k such that G admits a strong k-edge-coloring.
Strong edge-coloring was first studied by Fouquet and Jolivet [11, 12] for cubic planar graphs. By a greedy algorithm, it is easy to see that χ′s(G)≤ 2∆2−2∆+1 for any graph G of maximum degree ∆. Fouquet and Jolivet [11] established a Brooks type upper bound χ′s(G) ≤ 2∆2− 2∆, which is not true only for G = C5 as pointed out by Shiu and Tam [26]. The following conjecture was posed by Erd˝os and Neˇsetˇril [8, 9] and revised by Faudree, Gy´arf´as, Schelp and Tuza [10]:
Conjecture 1. If G is a graph of maximum degree ∆, then χ′s(G)≤ ∆2+⌊∆2⌋2. For graphs with maximum degree ∆ = 3, Conjecture 1 was verified by Andersen [1] and by Hor´ak, Qing and Trotter [15] independently. For ∆ = 4, while Conjecture 1 says that χ′s(G) ≤ 20, Hor´ak [14] obtained χ′s(G) ≤ 23 and Cranston [7] proved χ′s(G)≤ 22. Molloy and Reed [22] proved that for large ∆ every graph of maximum degree ∆ has χ′s(G) ≤ 1.998∆2 using probabilistic method. Mahdian [19] proved that for a C4-free graph G, χ′s(G) ≤ (2 + o(1))∆2/ ln ∆. Faudree, Gy´arf´as, Schelp and Tuza [10] proved that for graphs where all cycle lengths are multiples of four, χ′s(G)≤ ∆2. They mentioned that this result could probably be improved to a linear function of the maximum degree. Brualdi and Massey [2] improved the upper bound to χ′s(G) ≤ αβ for such graphs, where α and β are the maximum degrees of the respective partitions. Nakprasit [23] proved that if G is bipartite and the maximum degree of one partite set is at most 2, then χ′s(G)≤ 2∆. Chang and Narayanan [6]
proved that χ′s(G)≤ 8∆−6 for chordless graphs G. This settles the above question by Faudree, Gy´arf´as, Schelp and Tuza [10] in the positive, since graphs with cycle lengths divisible by 4 are chordless graphs. They also established that χ′s(G)≤ 10∆ − 10 for 2-degenerate graphs G.
Strong edge-coloring on planar graphs is also extensively studied in the litera- ture. Faudree, Gy´arf´as, Schelp and Tuza [10] asked whether χ′s(G)≤ 9 if G is cubic planar. If this upper bound is proved to be true, it would be the best possible.
Faudree, Gy´arf´as, Schelp and Tuza [10] used the Four-color theorem to show that χ′s(G) ≤ 4∆(G) + 4 for any planar graph G of maximum degree ∆. They also ex- hibited a planar graph G whose strong chromatic index is 4∆(G)− 4. Their proof also gives a consequence that χ′s(G) ≤ 3∆ for planar graphs G of girth at least 7.
Chang, Montassier, Pecher and Raspaud [5] further proved that χ′s(G) ≤ 2∆ − 1 for planar graphs G with large girth. Strong chromatic index for Halin graphs was first considered by Shiu, Lam and Tam [25] and then studied in [4, 16, 18, 26]. For trees G they obtained that χ′s(G) = σ(G), where
σ(G) := max
uv∈E(G)(dG(u) + dG(v)− 1) (1)
is an easy lower bound of χ′s(G), that is,
σ(G)≤ χ′s(G) for any graph G. (2)
An edge xy in a graph G is σ-tight if dG(x) + dG(y)− 1 = σ(G). Liao [17] studied cacti, which are graphs whose blocks are cycles or complete graphs of two vertices.
Notice that cacti are planar graphs and include trees. He established that for a cactus G, χ′s(G) = σ(G) if the length of any cycle is a multiple of 6, χ′s(G) ≤ σ(G) + 1 if the length of any cycle is even, and χ′s(G)≤ ⌊3σ(G)+12 ⌋ in general. For other results on strong edge-coloring, see [3, 13, 20, 21, 24, 27].
The purpose of this paper is to determine strong chromatic indices of cacti. The method is by means of jellyfish graphs to be introduced later. We first establish a decomposition theorem saying that the strong chromatic index of a graph is the maximum strong chromatic index of a block-jellyfish, which is a block together with edges with one vertex in the block and the other outside. Then we determine the strong chromatic index of a Cn-jellyfish, which is a graph obtained from the cycle Cn by attaching pendent edges to the cycle vertices.
2 Preliminary
For an integer n ≥ 3, the n-cycle is the graph Cn with vertex set V (Cn) = {v1, v2, . . . , vn} and edge set E(Cn) = {vivi+1: 1 ≤ i ≤ n}, where vn+1 = v1. More generally, when the indices of the vertices of an n-cycle are arithmetic expressions, they are considered to be taken modulo n.
A cut-vertex of a graph is a vertex whose removing results in a graph with more components than the old graph. A block of a graph is a maximal connected subgraph
without cut-vertices in itself. Any two blocks of a graph have at most one vertex in common, and if they meet at one vertex, then it is a cut-vertex. An end block is a block with exactly one cut-vertex. A block graph is a graph whose blocks are complete graphs. A cactus is a graph whose blocks are cycles or complete graphs of two vertices.
For a graph H, the H-jellyfish H(pv: v ∈ V (H)) is the graph obtained from H by adding pv new vertices adjacent to v for each vertex v in H. An edge joining a new vertex to v is called a pendent edge at v. For a block H of a graph G, any vertex u ∈ V (G) − V (H) is adjacent to at most one vertex v ∈ V (H), and if the vertex v exists then it is a cut-vertex of G. A block-jellyfish of a graph G is the H-jellyfish H′ for some block H of G, where the new vertices of H′ are all vertices of V (G)− V (H) having exactly one neighbor in V (H). A block-jellyfish is trivial if it is an H-jellyfish for an end block H which is K2, otherwise it is non-trivial.
Lemma 2. If H is a subgraph of G, then χ′s(H)≤ χ′s(G).
As any three consecutive edges in Cnuse different colors in a strong edge-coloring, the following lemma is an easy consequence of parity checking.
Proposition 3. If n≥ 3, then χ′s(Cn) = 5 for n = 5, χ′s(Cn) = 3 for n is a multiple of 3 and χ′s(Cn) = 4 otherwise.
Notice that a trivial block-jellyfish H1′ is a star; and if it is not a component, then it is a subgraph of a non-trivial block-jellyfish H2′. By Lemma 2, χ′s(H1′)≤ χ′s(H2′).
Theorem 4. Suppose G is a connected graph that is not a star. If G has exactly r non-trivial block-jellyfishes G1, G2, . . . , Gr, then χ′s(G) = max
1≤i≤rχ′s(Gi).
Proof. Since the graphs Gi are subgraphs of G, by Lemma 2, χ′s(G) ≥ max
1≤i≤rχ′s(Gi).
Next, we shall prove by induction on r that χ′s(G)≤ max
1≤i≤rχ′s(Gi). In the case where r = 1, G = G1 and so the inequality is clear. Assume r ≥ 2. Suppose the corre- sponding block of Gi in G is Hi. Then there is some Hi, say H1, which meets exactly one Hj at a cut-vertex of G. Let G′ be obtained from G by deleting G1 but keeping those vertices and edges in Gj undeleted. Then the non-trivial block-jellyfishes of G′ are exactly G2, G3, . . . , Gr. By the induction hypothesis, χ′s(G′) ≤ max
2≤i≤rχ′s(Gi).
Color G′ with χ′s(G′) colors. Since every two edges in E(G1) ∩ E(Gj) are adja- cent, meeting at the cut-vertex, we may assume that edges in E(G1)∩ E(Gj) are
colored by {1, 2, . . . , |E(G1)∩ E(Gj)|}. On the other hand, since d(e, e′) > 2 for any e ∈ E(G1)− E(Gj) and e′ ∈ E(G) − E(G1), we can color edges in E(G1) by {1, 2, . . . , χ′s(G1)}. Hence, we have colored G by max{χ′s(G′), χ′s(G1)} ≤ max
1≤i≤rχ′s(Gi) colors.
As an easy consequence, we have the following results for block graphs.
Corollary 5. If G is a block graph, then χ′s(G) = max{|E(H)|: H is a non-trivial block-jellyfish of G}.
Proof. This follows from Theorem 4 and the fact that any two edges in H are of distance within two.
Corollary 6. If G is a C3-jellyfish, then χ′s(G) =|E(G)|.
Proof. This follows from Corollary 5 and the fact that G is the only non-trivial block- jellyfish of itself.
Lemma 7. If G = H(pv: v ∈ V (H)) is an H-jellyfish such that {v : pv ̸= 0} ⊆ X ∪Y for two independent sets X and Y , then χ′s(G)≤ χ′s(H) + max{pu+ pv: u ∈ X, v ∈ Y, uv∈ E(H)}.
Proof. Let s = max{pu + pv: u ∈ X, v ∈ Y, uv ∈ E(H)}. For each vertex u ∈ X, color the pendent edges incident to u by {1, 2, . . . , pu}, and for each vertex v ∈ Y , color the pendent edges incident to v by {s − pv + 1, s− pv + 2, . . . , s}. We verify that the coloring is legal. In fact, if a pendent edge uu′ is within distance two from a pendent edge vv′, then uv∈ E(H). The assumption pu+ pv ≤ s gives pu < s−pv+ 1, so uu′ and vv′ are colored differently. We then use s + 1, s + 2, . . . , s + χ′s(H) to color the edges of H. These give a strong edge-coloring of G and the lemma follows.
Corollary 8. If G is a Cn-jellyfish with even n, then χ′s(G) ≤ σ(G) + χ′s(Cn)− 3.
Proof. Let X = {vi: i is odd} and Y = {vi: i is even}. The corollary follows from Lemma 7 and the fact that max
1≤i≤n(pi+ pi+1) = σ(G)− 3.
Corollary 9. ([17]) If G is a Cn-jellyfish with even n, then χ′s(G)≤ σ(G) + 1.
Proof. This follows from Corollary 8 and the fact that χ′s(Cn)≤ 4.
Corollary 10. If G is a C4-jellyfish, then χ′s(G) = σ(G) + 1.
Proof. By Corollary 9, χ′s(G)≤ σ(G) + 1. On the other hand, consider a cycle edge xy such that dG(x) + dG(y)− 1 = σ(G). Since the cycle edge not incident to x or y is within distance 2 from the edges incident to x or y, we have χ′s(G) ≥ σ(G) + 1, so χ′s(G) = σ(G) + 1.
Corollary 11. ([17]) If G is a Cn-jellyfish with n a multiple of 6, then χ′s(G) = σ(G).
Proof. This follows from Corollary 8 and the fact that χ′s(Cn) = 3.
Corollary 12. Suppose G is a Cn-jellyfish with dG(vj) = 2 for some j. If G ̸= C5, then χ′s(G)≤ σ(G) + 1. If n is a multiple of 3, then χ′s(G) = σ(G).
Proof. Without loss of generality, we may assume that j = n. Let X = {vi: i ̸= n and i is odd} and Y = {vi: i̸= n and i is even}.
Since max
1≤i≤n−1(pi + pi+1) = σ(G)− 3, by Lemma 7, χ′s(G) ≤ χ′s(Cn) + σ(G)− 3.
In the case where n̸= 5, this and χ′s(Cn)≤ 4 together imply that χ′s(G) ≤ σ(G) + 1.
In the case where n is a multiple of 3, this and χ′s(Cn) = 3 together imply that χ′s(G)≤ σ(G), so χ′s(G) = σ(G).
For the case of n = 5, consider the C5-jellyfish H = C5(min{pi, 1}: 1 ≤ i ≤ 5).
Notice that every cycle vertex of H has at most one pendent edge. Then χ′s(H)≤ 5, since we can color the edges of H with 5 colors by coloring the pendent edge at vi (if any) with the same color as the cycle edge vi+2vi+3, where the indices are taken modulo 5. Let p′i = pi − min{pi, 1} for 1 ≤ i ≤ 5. Notice that p5 = p′5 = 0 and
1≤i≤4max(p′i + p′i+1) ≤ σ(G) − 4, since there is at least one pi ̸= 0. Then G is the H- jellyfish H(p′i: 1 ≤ i ≤ 5), where the un-presented pu = 0 for all leaves u of H. By Lemma 7, χ′s(G)≤ max
1≤i≤4(p′i+ p′i+1) + χ′s(H)≤ σ(G) − 4 + 5 = σ(G) + 1.
3 Strong edge-coloring on cacti
The purpose of this section is to give the strong chromatic indices of cacti. Notice that a block-jellyfish of a cactus is either a K2-jellyfish or a Cn-jellyfish. The strong chromatic index of a K2-jellyfish is equal to its number of edges. So we only need to consider the case of Cn-jellyfish. Now suppose that G is a Cn-jellyfish. Notice that G = Cn(p1, p2, . . . , pn), where pi = dG(vi)− 2 for 1 ≤ i ≤ n. A rotation of a Cn-jellyfish G = Cn(p1, p2, . . . , pn) is a Cn-jellyfish G′ = Cn(p′1, p′2, . . . , p′n) with all
p′i = pi+r for a constant r, where the index in pi+r is taken modulo n. As we have the values for cycles in Proposition 3, we may only consider the case of σ(G)≥ 4.
Theorem 13. If G is a Cn-jellyfish of m edges with σ(G)≥ 4, then χ′s(G) =
m, if n = 3;
σ(G) + 1, if n = 4;
⌈⌊n/2⌋m ⌉, otherwise, if n is odd with all dG(vi) = d but (n, d)̸= (7, 3), or ⌈⌊n/2⌋m ⌉ ≥ σ(G) + 1;
σ(G) + 1, otherwise, if (n, d) = (7, 3) with all dG(vi) = d, or n̸≡ 0 (mod 3) such that up to rotation
dG(vi) = σ(G)− 1 for i ≡ 1 (mod 3) with 1 ≤ i ≤ 3⌊n3⌋ − 2, or (n, σ(G)) = (10, 4) with dG(vi) = 3 for all odd or all even i;
σ(G), otherwise.
To prove the main theorem, we first establish a sequence of lemmas as follows.
Lemma 14. If G is a Cn-jellyfish graph of m edges, then any color class of a strong edge-coloring has at most ⌊n/2⌋ edges and ⌈⌊n/2⌋m ⌉ ≤ χ′s(G).
Proof. We claim that there are at most ⌊n/2⌋ edges using the same color in a strong edge-coloring of G. For 1 ≤ i ≤ n, consider the set Ei consisting all edges incident to vi or vi+1 except the edge vi−1vi. Then for a fixed color c, each Ei contains at most one edge colored by c. As each edge of G appears in exactly two sets in E1, E2, . . . , En, there are at most⌊n/2⌋ edges using the color c. Hence ⌊n/2⌋m ≤ χ′s(G) and so ⌈⌊n/2⌋m ⌉ ≤ χ′s(G).
Lemma 15. If n is even or dG(vj) = 2 for some j, then ⌈⌊n/2⌋m ⌉ ≤ σ(G). If n is odd and dG(vi) = d for 1 ≤ i ≤ n, then ⌈⌊n/2⌋m ⌉ = σ(G) for 2 ≤ d ≤ (n + 1)/2,
⌈⌊n/2⌋m ⌉ = σ(G) + 1 for (n + 3)/2 ≤ d ≤ n and ⌈⌊n/2⌋m ⌉ ≥ σ(G) + 2 for d ≥ n + 1.
Proof. If n is even or dG(vj) = 2 for some j, say j = n, then m−1 ≤∑⌊n/2⌋
i=1 (dG(v2i−1)+
dG(v2i)− 2) ≤ ⌊n/2⌋(σ(G) − 1), so ⌊n/2⌋m ≤ σ(G). If n is odd and dG(vi) = d for 1 ≤ i ≤ n, then ⌈⌊n/2⌋m ⌉ = 2d − 2 + ⌈2dn−1−2⌉ = σ(G) − 1 + ⌈2dn−1−2⌉, which is σ(G) for 2 ≤ d ≤ (n + 1)/2, is σ(G) + 1 for (n + 3)/2 ≤ d ≤ n and is at least σ(G) + 2 for d≥ n + 1.
Lemma 16. If G is a Cn-jellyfish with dG(vi) = d≥ 3 for 1 ≤ i ≤ n, then
χ′s(G) =
σ(G) + 1, if n = 4 or (n, d) = (7, 3);
σ(G), if n≥ 6 is even;
⌈⌊n/2⌋m ⌉, if n≥ 3 is odd but (n, d) ̸= (7, 3).
Proof. Case 1. n = 4. In this case, the lemma follows from Corollary 10.
Case 2. (n, d) = (7, 3). In this case, an induced matching has at most 3 edges by Lemma 14. If χ′s(G) ≤ 5, then there are at least two color classes containing 2 cycle edges, each of which must be precisely of size 2. Then all color classes contain at most 13 edges, contradicting to the fact that G has 14 edges. Hence χ′s(G) ≥ 6.
Figure 1 gives a strong 6-edge-coloring of G, so χ′s(G) = 6 = σ(G) + 1.
u 3 u 2 u 1 u 6u
u5
u6
u5
4 3
u 1 u 2 u
u
5
u
6
u
4
Figure 1: A strong 6-edge-coloring of G.
Case 3. n ≥ 6 is even. In this case, by Corollary 11, we only need to consider the case of n̸≡ 0 (mod 3). For 1 ≤ i ≤ n, let ei = vivi+1and fi,1, fi,2, . . . , fi,d−2 be the pendent edges at vi. The lemma follows from the fact that we may partition E(G) into σ(G) = 2d− 1 induced matchings as follows:
for n≡ 2 (mod 3),
M1 ={f1,1, e3} ∪ {ei: 6≤ i ≤ n, i ≡ 0 (mod 3)}, M2 ={f3,1, e5} ∪ {ei: 6≤ i ≤ n, i ≡ 2 (mod 3)}, M3 ={f5,1, e2} ∪ {ei: 6≤ i ≤ n, i ≡ 1 (mod 3)}, M4 ={e1, e4} ∪ {fi,1: 7≤ i ≤ n, i ≡ 1 (mod 2)}, M5 ={fi,1: 2≤ i ≤ n, i ≡ 0 (mod 2)};
for n≡ 1 (mod 3),
M1 ={f1,1, f6,1, e3} ∪ {ei: 6≤ i ≤ n, i ≡ 2 (mod 3)}, M2 ={f2,1, f4,1} ∪ {ei: 6≤ i ≤ n, i ≡ 0 (mod 3)}, M3 ={f3,1, f5,1} ∪ {ei: 6≤ i ≤ n, i ≡ 1 (mod 3)}, M4 ={e1, e4} ∪ {fi,1: 7≤ i ≤ n, i ≡ 1 (mod 2)}, M5 ={e2, e5} ∪ {fi,1: 7≤ i ≤ n, i ≡ 0 (mod 2)};
for 2≤ j ≤ d − 2,
{ M2j+2 ={fi,j: 1≤ i ≤ n, i ≡ 1 (mod 2)}, M2j+3 ={fi,j: 1≤ i ≤ n, i ≡ 0 (mod 2)}.
Case 4. n ≥ 3 is odd but (n, d) ̸= (7, 3). In this case, ⌊n/2⌋ = (n − 1)/2, m = n(d− 1) and ⌊n/2⌋m = 2n(d− 1)/(n − 1). By Lemma 14, χ′s(G) ≥ ⌈⌊n/2⌋m ⌉. We shall show the upper bound by considering two subcases.
Case 4-1. 3 ≤ d ≤ (n + 1)/2. We will show that χ′s(G) ≤ 2d − 1 and since 2d− 1 < ⌊n/2⌋m + 1, χ′s(G) =⌈⌊n/2⌋m ⌉. For any integer t and odd q with 1 ≤ q ≤ n/3, let
I(t, q) ={et+3, et+6, et+9, . . . , et+3q} ∪ {ft+3q+3, ft+3q+5, ft+3q+7, . . . , ft+n+1}, where the indices are taken modulo n and we denote every pendent edge at vi by fi for simplicity. This is an induced matching containing q cycle edges and (n− 3q)/2 pendent edges. Since 5 ≤ 2d − 1 ≤ n, n is odd and (n, d) ̸= (7, 3), we can write n as a sum of 2d− 1 odd numbers q1, q2, . . . , q2d−1, each of which is no more than n/3.
This can be done by choosing qi’s such that the gap between the maximum and the minimum is at most 2. Let Qi =∑i
j=13qj for 0≤ i ≤ 2d − 1.
In the case where n is not a multiple of 3, we claim that the induced match- ings I(Qi−1, qi), for 1 ≤ i ≤ 2d − 1, partition E(G). First, the cycle edges used are e3, e6, e9, . . . , e3q1; e3q1+3, e3q1+6, e3q1+9, . . . , e3q1+3q2; . . . ; eQ2d−2+3, eQ2d−2+6, eQ2d−2+9, . . . , eQ2d−2+3q2d−1 = e3n, which cover each cycle edge exactly once as n is not a multiple of 3.
Secondly, the pendent edges used, viewing backward, are fn+1, fn−1, fn−3, . . . , f3q1+3; f3q1+1, f3q1−1, f3q1−3, . . . , f3q1+3q2+3; . . . ; fQ2d−2+1, fQ2d−2−1, fQ2d−2−3, . . . , fQ2d−2+3q2d−1+3
= fn+3, which cover pendent edges at each cycle vertex exactly 1n∑2d−1
i=1 (n− 3qi)/2 = d− 2 times.
In the case where n is a multiple of 3, we modify the above arguments as follows.
We may assume that n ≥ 9 as the case for n = 3 follows from Corollary 6. In this case, we may choose q1, q2, . . . , q2d−1 so that they can be divided into 3 parts, each summing up to n/3. This can be done by first choosing q1, q2, q3 to be n/3, n/3, n/3, and then properly shifting 1’s from them to other qi’s. We adopt similar arguments as above for the 3 parts separately, but consider rather Qi = r +∑i
j=13qj for the rth part, 1 ≤ r ≤ 3. The induced matchings in part r cover all the n/3 cycle edges ei with i ≡ r (mod 3), and the pendant edges fr+n+1, fr+n−1, fr+n−3, . . . , fr+3(3n)+3, with a total number of a multiple of n. Similarly, pendant edges at each cycle vertex are also covered d− 2 times.
Case 4-2. d > (n + 1)/2. In this case, we partition the edges of G into two parts: the first part consists of the cycle edges together with (n − 3)/2 pendent edges at each cycle vertex, and the second part consists of d− (n + 1)/2 pendent
edges at each cycle vertex. The first part has m1 = n(n− 1)/2 edges. By Case 4-1, it can be partitioned into n induced matchings. Next, we order the pendant edges in the second part as h1, h2, . . . , hm−m1, where hj is a pendant edge at cycle vertex vi with i ≡ 2j − 1 (mod n). Notice that for any integer t and any integer r≤ (n − 1)/2, the set {ht+1, ht+2, . . . ht+r} is an induced matching. Hence the second part can be partitioned into ⌈(nm−1)/2−m1 ⌉ induced matchings. Totally, the edges of G can be partitioned into ⌈(n−1)/2m ⌉ as desired.
We now consider the case where dG(vj) = 2 for some vj, say vn. By Corollary 12, χ′s(G) = σ(G) or σ(G) + 1.
Lemma 17. If n ̸≡ 0 (mod 3) and G is a Cn-jellyfish such that dG(vi) = σ(G)− 1 for i≡ 1 (mod 3) and 1 ≤ i ≤ 3⌊n/3⌋ − 2, then χ′s(G) = σ(G) + 1.
Proof. First, the assumption gives that dG(vj) = 2 for j = n or j ̸≡ 1 (mod 3) with 1 ≤ j ≤ 3⌊n/3⌋ − 1. By Corollary 12, χ′s(G) ≤ σ(G) + 1. Suppose to the contrary that G had a strong edge-coloring using σ(G) colors. Then for each i ≡ 1 (mod 3) with 1 ≤ i ≤ 3⌊n/3⌋ − 2, the σ(G) − 3 pendent edges at vi, ei−1, ei, together with ei−2 (respectively, ei+1) would use all the σ(G) colors. It follows that ei−2 and ei+1 would use the same color. Hence en−1, e2, e5, . . . , e3⌊n/3⌋−1 would all use the same color. Since n̸≡ 0 (mod 3), en−1 and e3⌊n/3⌋−1 are two distinct edges within distance two, which leads to a contradiction.
Lemma 18. If G is a C10-jellyfish such that dG(vi) = σ(G)− 1 = 3 for all odd i, then χ′s(G) = σ(G) + 1 = 5.
Proof. First, the assumption gives that dG(vj) = 2 for all even j. By Corollary 12, χ′s(G)≤ σ(G) + 1. Suppose to the contrary that G had a strong edge-coloring using σ(G) = 4 colors. Then for each odd i, the σ(G)− 3 = 1 pendent edge at vi, ei−1, ei, together with ei−2 (respectively, ei+1) would use all the σ(G) colors. This gives that ei−2 and ei+1 use the same color. Since we only had 4 colors for the 10 cycle edges, there would be one color used for at least 3 edges. But a color should appear in a pair of edges as shown above. This color would then be used for at least 4 cycle edges, which is impossible.
Lemma 19. If G is a Cn-jellyfish with σ(G) = 4, then χ′s(G) = σ(G) except that χ′s(G) = σ(G) + 1 when, up to rotation, n ̸≡ 0 (mod 3) such that dG(vi) = 3 for i≡ 1 (mod 3) with 1 ≤ i ≤ 3⌊n3⌋ − 2 or n = 10 such that dG(vi) = 3 for all odd i.
Proof. If n is a multiple of 3, then the lemma follows from Corollary 12. Now assume that n̸≡ 0 (mod 3). The exceptional cases follow from Lemmas 17 and 18.
Up to rotation, we may assume that 1 = i1 < i2 < . . . < is are all the indices for which vir is of degree 3. For 1 ≤ r ≤ s, the path Pr from vir to vir+1 consists of nr = ir+1− ir cycle edges, where is+1 = n + 1. Using this notion, the Cn-jellyfish G is completely determined by the sequence n1, n2, . . . , ns. Notice that the first exceptional case is the same as that all nr = 3 except exactly one nr ∈ {2, 4, 5} or exactly two consecutive nr = 2, and the second exceptional case is the same as that n = 10 and all nr = 2. We consider cases other than the two exceptional cases. Since the cases for n = 4, 5 are included in the first exceptional case, and 6 is a multiple of 3, we may assume n≥ 7. The aim is to find a strong 4-edge-coloring for G. By adding suitable pendent edges, we may assume that all nr ∈ {2, 3} and there are two non-consecutive nr = 2.
If there is at least one nr = 3, then up to rotation we may assume that ns = 2, and there exists some t ≤ s − 1 such that nr = 3 for all 1 ≤ r ≤ t and nt+1 = 2.
Otherwise, if all nr = 2, then s ≥ 4, s ̸= 5, and we choose t = 0. We define an edge-coloring on cycle edges first as follows.
@@
@u
v1u
e1
3 3
u e2
1 1
u e3
2 2
u e4
3 3
u e5
1 1
u. . . u
e3t−3
2 2
ue3t−2
3 3
ue3t−1
1 1
ue3t
2 2
u u
u
v3t+1
en 4
2 u
en−1 1 1
u
vn−1uen−2 2 3
uen−3 4 2
uen−4 3 4
uen−5 2 3
uen−6 4 2
u. . . ue3t+6 3 3
ue3t+5 2 2
ue3t+4 4 4
ue3t+3 3 3
uv3t+3
@@@
e3t+1 4 4
u
e3t+2 1 1
Figure 2: Inner labels are for n≡ 1 (mod 3) and outer labels are for n ≡ 2 (mod 3).
Notice that in the coloring as shown in Figure 2, the edges in the upper path e1, e2, e3, . . . , e3t−2, e3t−1, e3t are colored by 3, 1, 2 periodically; the edges in the lower path e3t+3, e3t+4, e3t+5, . . . , en−3−x, en−2−x, en−1−x are colored by 3, 4, 2 periodically, where x = (n mod 3). These colors for the cycle edges satisfy the following two conditions.
(i) Any two distinct cycle edges within distance two receive distinct colors.
(ii) The two cycle edges with distance exactly two from a pendent edge receive a same color.
By (ii), the four cycle edges within distance two from a pendent edges use only 3 colors. Hence we may color any pendent edge with the remaining color to form a strong 4-edge-coloring of G.
Lemma 20. If G is a Cn-jellyfish with σ(G) ≥ 4 and dG(vj) = 2 for some j, then χ′s(G) = σ(G) except that χ′s(G) = σ(G) + 1 when, up to rotation, n ̸≡ 0 (mod 3) such that dG(vi) = σ(G)− 1 for i ≡ 1 (mod 3) with 1 ≤ i ≤ 3⌊n3⌋ − 2 or n = 10 such that dG(vi) = σ(G)− 1 = 3 for all odd i.
Proof. The exceptional cases follow from Lemmas 17 and 18. We shall prove the lemma by induction on σ(G). The case of σ(G) = 4 follows from Lemma 19. Now assume that σ(G)≥ 5.
A run is a maximal sequence vi, vi+1, . . . , vi+j of cycle vertices in which every vertex is of degree at least 3. The even-half (respectively, odd-half) of the run is the vertices vi+r with 0 ≤ r ≤ j and r even (respectively, odd). Notice that an even- half of a run is always non-empty, while an odd-half is empty if and only if j = 0.
Consider a Cn-jellyfish G′ obtained from G by deleting a pendent edge at each vertex of exactly one of the even-half or the odd-half of each run. Then σ(G) = σ(G′) + 1 and χ′s(G)≤ χ′s(G′) + 1 as the deleted edges form an induced matching.
Suppose that G′ is not in the exceptional cases. By the induction hypothesis, χ′s(G′) = σ(G′). Then χ′s(G) ≤ χ′s(G′) + 1 = σ(G′) + 1 = σ(G), so χ′s(G) = σ(G).
Now we may assume that G′ is in the exceptional cases. If there is a run of length one in G′ obtained from some run of length not one in G, then we change to delete the other half of this run in G and obtain a new G′ which is not in the exceptional cases. Now every run of length one in G′ is obtained from a run of length one in G, and since G′ is in the exceptional cases but not G, it must be that n = 10 and dG(vi) = σ(G)− 1 = 4 for all odd i. Then χ′s(G) = σ(G) = 5 as shown in Figure 3.
u 3 u 1 u 2 u 5 u
5u A
AAu4 4u A
AAu3
2 1
u 4 u 3 u 5 u 4 u
AA Au
5
u
1 AuAA
1
u
2 AuAA
2
u
3
Figure 3: The C10-jellyfish G with dG(vi) = σ(G)− 1 = 4 for all odd i.
Having the above lemmas established, we are now ready to prove Theorem 13.
For the case of n = 3, the theorem follows from Corollary 6. For the case of n = 4, the theorem follows from Corollary 10. Now we may assume that n≥ 5.
If dG(vj) = 2 for some j, then⌈⌊n/2⌋m ⌉ ≤ σ(G) by Lemma 15, so the third case of the theorem does not happen. The theorem then follows from Lemma 20.
We now consider the case where dG(vi)≥ 3 for all i. There are two subcases to be considered depending on the parity of n.
We first consider the subcase where n is even. By Lemma 15, ⌈⌊n/2⌋m ⌉ ≤ σ(G), so the third case of the theorem does not happen. We then only need to prove that χ′s(G) = σ(G). Let H be the Cn-jellyfish with each cycle vertex vi of degree 3. By Lemma 16, χ′s(H) = 5. Then G is obtained from H by adding pvi = dG(vi)−3 pendent edges at vi for 1≤ i ≤ n. Let X = {vi: 1≤ i ≤ n and i odd} and Y = {vi: 1≤ i ≤ n and i even}. Then max{pu+ pv: u∈ X, v ∈ Y, uv ∈ E(H)} = σ(G) − 5. By Lemma 7, χ′s(G)≤ χ′s(H) + max{pu+ pv: u∈ X, v ∈ Y, uv ∈ E(H)} ≤ 5 + σ(G) − 5 = σ(G) and so χ′s(G) = σ(G).
Next we consider the second subcase when n is odd. If all cycle edges vivi+1 are tight, then dG(vi) + dG(vi+1)− 1 = dG(vi+1) + dG(vi+2)− 1 and so dG(vi) = dG(vi+2) for all i. Since n is odd, all dG(vi) are equal.
Suppose, up to rotation, that vnv1 is a non-tight edge. Consider the Cn-jellyfish graph G1 obtained from G0 := G by deleting one pendent edge at vi for all even i. Then σ(G1) = σ(G0)− 1 and G1 has m1 = m− ⌊n/2⌋ edges. Since we can use one color for the deleted edges, χ′s(G0) ≤ χ′s(G1) + 1. Repeating the same process gives that there is an integer s≥ 0 and Cn-jellyfish graphs G0, G1, . . . , Gs such that σ(Gr) ≥ 4, σ(Gr) = σ(G)− r, Gr has mr = m− r⌊n/2⌋ edges, χ′s(G)≤ χ′s(Gr) + r for 0 ≤ r ≤ s, and either dGs(vj) = 2 for some j or else dGs(vi) is a constant ds for all i.
For the former case, Gs−1 has the property that all cycle vertices have degree at least 3. But after deleting (n− 1)/2 pendent edges, the resulting graph Gs has some cycle vertex vjwith degree 2. It then must be the case that Gsis not in the exceptional cases in Lemma 20. Hence χ′s(Gs) = σ(Gs) and χ′s(G) ≤ χ′s(Gs) + s = σ(Gs) + s = σ(G). By Lemma 15, ⌊n/2⌋ms ≤ σ(Gs), so ⌊n/2⌋m = ⌊n/2⌋ms + s ≤ σ(Gs) + s = σ(G). It follows that G fits the fifth case.
Now we may assume that dGs(vi) is a constant ds for all i. If (n, ds)̸= (7, 3), then
by Lemma 16, χ′s(Gs) =⌈⌊n/2⌋ms ⌉ and so χ′s(G)≤ χ′s(Gs) + s =⌈⌊n/2⌋ms ⌉ + s = ⌈⌊n/2⌋m ⌉.
By Lemma 14, χ′s(G) = ⌈⌊n/2⌋m ⌉. If (n, ds) = (7, 3), then Gs−1 must be the graph as in Figure 4 from which we conclude that χ′s(Gs−1) = σ(Gs−1). Then χ′s(G) ≤ χ′s(Gs−1) + s− 1 = σ(Gs−1) + s− 1 = σ(G). Notice that ⌈⌊n/2⌋ms ⌉ = 5 = σ(Gs) and
⌈⌊n/2⌋m ⌉ = ⌈⌊n/2⌋ms ⌉ + s = σ(Gs) + s = σ(G), so G fits the fifth case.
u 5 u 3 u 1 u
1u A AAu2
u4
u2 5u A
AAu4
6 6
u 4 u 2 u
u
3 AuAA
5
u
1
u
3
Figure 4: The graph Gs−1 for (n, ds) = (7, 3).
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