On the word-representability of $K_m$-$K_n$ graphs
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915454540316672 |
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| author | Chen, Herman Z. Q. Hameed, Humaira Kitaev, Sergey |
| author_facet | Chen, Herman Z. Q. Hameed, Humaira Kitaev, Sergey |
| contents | Word-representable graphs are a class of graphs that can be represented by words, where edges and non-edges are determined by the alternation of letters in those words. Several papers in the literature have explored the word-representability of split graphs, in which the vertices can be partitioned into a clique and an independent set. In this paper, we initiate the study of the word-representability of graphs in which the vertices can be partitioned into two cliques. We provide a complete characterization of such word-representable graphs in terms of forbidden subgraphs when one of the cliques has a size of at most four. In particular, if one of the cliques is of size four, we prove that there are seven minimal non-word-representable graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_15177 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the word-representability of $K_m$-$K_n$ graphs Chen, Herman Z. Q. Hameed, Humaira Kitaev, Sergey Combinatorics Word-representable graphs are a class of graphs that can be represented by words, where edges and non-edges are determined by the alternation of letters in those words. Several papers in the literature have explored the word-representability of split graphs, in which the vertices can be partitioned into a clique and an independent set. In this paper, we initiate the study of the word-representability of graphs in which the vertices can be partitioned into two cliques. We provide a complete characterization of such word-representable graphs in terms of forbidden subgraphs when one of the cliques has a size of at most four. In particular, if one of the cliques is of size four, we prove that there are seven minimal non-word-representable graphs. |
| title | On the word-representability of $K_m$-$K_n$ graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.15177 |