A note on the clique number of complete $k$-partite graphs

In this note, we show that a complete $k$-partite graph is the only graph with clique number $k$ among all degree-equivalent simple graphs. This result gives a lower bound on the clique number, which is sharper than existing bounds on a large family of graphs.

💡 Research Summary

The paper investigates the relationship between degree sequences and the clique number of graphs, focusing on complete k‑partite graphs. After recalling basic graph‑theoretic notions (degree, degree sequence, degree‑equivalence, complement, clique number ω, independence number α), the author emphasizes that a complete k‑partite graph K_{a₁,…,a_k} and its complement (a disjoint union of cliques K_{a₁}∪…∪K_{a_k}) are dual structures: cliques in one correspond to independent sets in the other.

Turán’s theorem is invoked to remind the reader that among all K_{k+1}‑free graphs on n vertices, the Turán graph T(n,k) (the balanced complete k‑partite graph) maximizes the number of edges. From this classic result, Corollary 1 follows: if a graph G has the same degree sequence as K_{a,…,a} (all parts equal) and ω(G)=k, then G must be exactly K_{a,…,a}.

The main contribution, Theorem 1, generalizes this to arbitrary part sizes. It states: for any graph G degree‑equivalent to K_{a₁,…,a_k}, the equality ω(G)=k holds if and only if G is precisely the complete k‑partite graph K_{a₁,…,a_k}. The proof is carried out via the complement: the equivalent statement (Corollary 3) asserts that if G is degree‑equivalent to a disjoint union of cliques K_{a₁}∪…∪K_{a_k} but is not exactly that union, then its independence number satisfies α(G)≥k+1.

The proof proceeds by contradiction and induction. Assuming G has no clique components, the part sizes are ordered a₁≤…≤a_k, and the vertices are partitioned into sets S_i each containing a_i vertices of degree a_i−1. For the smallest block of equal sizes (a₁=…=a_c<a_{c+1}), the subgraph G_c induced by the first c blocks is examined. Using elementary set‑theoretic inequalities, it is shown that any maximal independent set in G_c must have size at least c+1; otherwise a vertex not adjacent to the set would exist, contradicting maximality. Moreover, the structure of neighborhoods forces the existence of a clique component of size a₁, which contradicts the assumption that G has no such component. Hence α(G_c)≥c+1.

The inductive step adds one more block at a time, showing that if an independent set of size c+j+1 exists in G_{c+j}, then a vertex outside this set is not adjacent to all its members (by a counting argument using the degree bounds a_i−1). This vertex can be added, increasing the independent set size to c+j+2. Repeating until the whole graph G=G_k yields α(G)≥k+1, establishing the desired bound.

The paper also discusses algorithmic aspects. Proposition 1 shows that checking whether a graph is a complete k‑partite graph, its complement, or degree‑equivalent to either can be done in O(m + n log n) time, where n and m are the numbers of vertices and edges.

Proposition 2 proves that, despite the simple degree‑sequence condition, computing α(G) or ω(G) for graphs degree‑equivalent to a disjoint union of cliques (or a complete k‑partite graph) remains NP‑complete. The reduction is from the independence‑number problem on cubic graphs, constructing four disjoint copies of a cubic graph to obtain a graph whose degree sequence matches that of a union of cliques; the independence number scales linearly, preserving hardness. Since the complement operation swaps independence and clique numbers, the same hardness holds for the clique‑number problem on degree‑equivalent complete multipartite graphs.

Finally, the author compares the new bounds with several classic lower bounds on α(G) and ω(G): the Caro‑Wei bound, Turán’s bound, Hansen‑Zheng, Myer‑Liu, and Edwards‑Elphick. For graphs satisfying the degree‑equivalence condition, these known bounds typically guarantee only α(G)≥k (or ω(G)≥k), whereas Corollaries 3 and 4 guarantee α(G)≥k+1 (or ω(G)≥k+1), which is sharp as demonstrated by the example graph in Figure 1. Thus, the paper provides a stronger, degree‑sequence‑based criterion for the existence of a (k+1)‑clique (or independent set) in a broad family of graphs, extending Turán’s extremal result beyond the balanced case and offering a useful tool for graph‑theoretic analysis where exact computation of ω or α is computationally infeasible.