Semistrong Edge Coloring in Graph Theory

Updated 28 September 2025

Semistrong edge coloring is a refined proper edge coloring where each color class forms a matching with at least one pendant vertex, providing a middle ground between proper and strong colorings.
It offers tight upper bounds for various classes such as general graphs, complete bipartite graphs, subcubic graphs, and planar graphs, thus enhancing applications in network modeling.
Algorithmic analysis reveals efficient dynamic programming solutions for trees and NP-completeness for k ≥ 3, underlining its significance in resource allocation and scheduling.

A semistrong edge coloring is a refinement of proper edge coloring in which each color class forms not just a matching but a matching that is “locally sparse” in a well-defined sense: every edge in a color class is incident to a vertex that is a leaf (degree one) in the subgraph induced by the color class. This definition creates an intermediate concept between proper edge coloring and strong edge coloring (where color classes are induced matchings) and has important applications in graph theory and network modeling.

1. Fundamental Definitions and Notation

Let $G = (V, E)$ be a finite simple graph. A matching $M \subseteq E$ is a set of pairwise nonadjacent edges. The subgraph induced by all endvertices of $M$ is denoted $G[V(M)]$ . A matching $M$ is semistrong if every edge $e \in M$ has at least one endvertex of degree one in $G[V(M)]$ —that is, each edge is incident with a pendant vertex in $G[V(M)]$ .

A semistrong edge coloring of $G$ is a proper edge coloring where every color class forms a semistrong matching. The semistrong chromatic index, $\chi_{ss}'(G)$ , is the minimum number of colors such a coloring can use. The following chain of inequalities holds for any graph $G$ : $\chi'(G) \le a'(G) \le \chi'_{ur}(G) \le \chi_{ss}'(G) \le \chi_s'(G)$ where:

$\chi'(G)$ = chromatic index (proper edge coloring),
$a'(G)$ = acyclic chromatic index,
$\chi'_{ur}(G)$ = uniquely restricted chromatic index,
$\chi_{ss}'(G)$ = semistrong chromatic index,
$\chi_s'(G)$ = strong chromatic index [(Lužar et al., 2022, Lin et al., 26 Dec 2024, Lin et al., 21 Sep 2025)].

In the specific case of a strong edge coloring, each color class forms an induced matching (no two edges are connected by a path of length one in $G[V(M)]$ ), a strictly stronger requirement than for semistrong matching. In particular, induced matching implies that all vertices in $G[V(M)]$ are of degree one, while the semistrong matching only requires each edge to possess at least one such endpoint.

2. Main Theoretical Results and Upper Bounds

The semistrong chromatic index is bounded above in general by quadratic functions of the maximum degree, but much sharper bounds exist for special classes:

General Graphs: For a graph $G$ with maximum degree $\Delta$ ,

$\chi_{ss}'(G) \le \Delta^2$

[(Lužar et al., 2022)]. More recently, this bound was improved to $\chi_{ss}'(G) \le \Delta^2 - 1$ for all connected graphs except the complete bipartite graph $K_{\Delta,\Delta}$ and cycles of length 7 [(Lin et al., 2023)].

Complete Bipartite Graphs: $\chi_{ss}'(K_{\Delta,\Delta}) = \Delta^2$ .
Cycles and Paths: For graphs with $\Delta = 2$ (cycles and paths), $\chi_{ss}'(G) \le 3$ , except for some cases (e.g., $C_7$ requires $4$ colors).
Subcubic Graphs ( $\Delta = 3$ ): Every connected subcubic graph except $K_{3,3}$ satisfies $\chi_{ss}'(G) \le 8$ , and this is tight (the 5-prism achieves the bound). $K_{3,3}$ requires nine colors [(Lužar et al., 2022, Lin et al., 2023)].
Planar Graphs: For any planar graph $G$ with maximum degree $\Delta$ , if the maximum average degree (mad) is less than $14/5$, then $\chi_{ss}'(G) \le 2\Delta + 4$ ; if mad $< 8/3$ , then $\chi_{ss}'(G) \le 2\Delta + 2$ . For planar graphs with girth at least $7$ (or $8$), these bounds are exact [(Lin et al., 26 Dec 2024)].
Examples of Tightness: There exist planar graphs for which $\chi_{ss}'(G) = 2\Delta + 4$ , showing the tightness of the $2\Delta + 4$ bound for these classes.
Algorithmic Results for Trees: Every tree $T$ with maximum degree $\Delta$ satisfies $\chi_{ss}'(T) \in \{\Delta, \Delta+1\}$ [(Lin et al., 21 Sep 2025)].
Computational Complexity: Determining if $\chi_{ss}'(G) \le k$ is polynomial-time solvable for $k \le 2$ , but becomes NP-complete for $k \ge 3$ [(Lin et al., 21 Sep 2025)].

3. Relationships to Other Coloring Concepts

Semistrong edge coloring sits between the classical and strong edge coloring concepts and relates closely to other intermediates:

For a matching to be induced, all degrees in $G[V(M)]$ must be one; for uniquely restricted, $M$ must be the unique perfect matching of $G[V(M)]$ .
Semistrong matchings are always uniquely restricted, and induced matchings are always semistrong. Thus,

$\text{Induced matching} \implies \text{Semistrong matching} \implies \text{Uniquely restricted matching} \implies \text{Matching}$

Packing Edge Colorings: The semistrong condition corresponds to $(0,1)$ -relaxed strong edge coloring, where for each edge $e$ , at most 0 other edge at distance one and at most one edge at distance two can share its color. Intermediate colorings such as $(1^a, 2^b)$ -packing edge colorings, which decompose the edge set into a mix of matchings (distance at least two) and induced matchings (distance at least three), are also studied and connected with semistrong edge coloring [(Hocquard et al., 2020)].
Simultaneous Edge Coloring: The notion of $p$ -simultaneous edge coloring, where multiple proper edge colorings are required to have matching color-sets incident at each vertex, provides a strict combinatorial model. A plausible implication is that methods for decomposing simultaneous edge colorings—such as those based on Latin trades, CDCs, and graph products—may adapt to the semistrong setting [(Gh. et al., 2013)].

4. Algorithmic and Complexity Considerations

General Case: The decision version—does $G$ have a semistrong $k$ -edge coloring?—is in P for $k \leq 2$ but is NP-complete for $k \geq 3$ [(Lin et al., 21 Sep 2025)]. The reduction uses the known hardness of coloring regular graphs and properties of semistrong matchings.
Trees: A dynamic programming approach in $O(\Delta^6 n)$ time exactly decides the semistrong chromatic index of a tree $T$ with $n$ vertices and maximum degree $\Delta$ [(Lin et al., 21 Sep 2025)]. The algorithm computes and merges “coloring profiles” at each subtree rooted at a vertex, using “vertical expansion” and “horizontal merging” steps that count configurations of pending and incident colors per child vertex.
Planar Graphs and Discharging Techniques: For planar graphs, structural sparsity (expressed in maximum average degree or girth) allows for tighter bounds and inductive/dicharging arguments.

5. Applications and Modeling Significance

Semistrong edge coloring appears in multiple applied and theoretical domains:

Wireless Network Scheduling: The problem models conflict-averse assignment of communication slots. Vertices are devices; edges are communication links; time slots (colors) are assigned so that in each slot, the corresponding assignment is a matching and every connection (edge) is “safe”—i.e., at least one endpoint is a leaf, reducing potential interference [(Lin et al., 21 Sep 2025)].
Efficient Resource Allocation: Semistrong matchings guarantee configurations where each scheduled “task” (edge) has isolation at one endpoint, aiding scenarios where exclusive access at one node is sufficient.
Graph Theoretic Benchmarks: The semistrong chromatic index is used to estimate or bound the minimum resources needed under “relaxed yet safe” coloring constraints.
Complexity Benchmarking: The rapid rise in computational complexity between $k = 2$ and $k = 3$ underscores the challenge of general coloring problems with even mildly enhanced local conditions.

6. Open Problems and Research Directions

Several directions remain open or are explicitly conjectured:

Tightening General Upper Bounds: Is $\chi_{ss}'(G) \leq \Delta^2 - \Delta + 1$ for all graphs with sufficiently large $\Delta$ ? Construction of graphs achieving this lower bound is possible, but no general proof yet exists [(Lin et al., 2023)].
Planar Graphs: Does every planar graph satisfy $\chi_{ss}'(G) \leq 2\Delta + 4$ ? This is conjectured and demonstrated for certain graph families and shown to be sharp for some $(\Delta, g)$ pairs [(Lin et al., 26 Dec 2024)].
Graph Classes: Can algorithmic or structural results be extended to broader classes such as bipartite graphs, $C_4$ -free graphs, or partial $k$ -trees?
Relationship to Other Parameters: Further investigation is needed into how the semistrong chromatic index relates quantitatively and structurally to the acyclic, uniquely restricted, and strong chromatic indices.
Algorithmic Extensions: The development of efficient algorithms beyond trees, such as approximation algorithms for general graphs or specialized methods for sparse graphs, remains an open challenge.
Connections to Simultaneous and Signed Colorings: Analysis of the translation between $p$ -simultaneous colorings, signed edge colorings, and semistrong colorings in various graph classes could yield unified frameworks and algorithmic tools [(Gh. et al., 2013, Behr, 2018)].

Coloring Notion	Local Structure per Color Class	Chromatic Index Notation
Proper (matching)	Matching	$\chi'(G)$
Uniquely restricted	Unique perfect matching in induced subg.	$\chi'_{ur}(G)$
Semistrong	Every edge has a degree-1 endpoint	$\chi_{ss}'(G)$
Strong (induced)	Each class is induced matching	$\chi_s'(G)$

This table organizes the hierarchy and relative strength of constraints among edge coloring parameters.

In conclusion, semistrong edge coloring formalizes a robust middle ground between the classical and strong chromatic indices, supported by tight extremal bounds, precise computational complexity thresholds, significant connections to resource allocation models and wireless scheduling, and a wealth of open problems related to both structure and computation. Recent advances have sharply delineated the achievable bounds and clarified the structural and algorithmic scope of this parameter, establishing it as a central notion in modern algorithmic graph theory [(Lužar et al., 2022, Lin et al., 2023, Lin et al., 26 Dec 2024, Lin et al., 21 Sep 2025)].