Words and Graphs by Sergey Kitaev, Vadim Lozin

By Sergey Kitaev, Vadim Lozin

This is the 1st complete advent to the idea of word-representable graphs, a generalization of a number of classical periods of graphs, and a brand new subject in discrete mathematics.

After broad introductory chapters that specify the context and consolidate the state-of-the-art during this box, together with a bankruptcy on hereditary periods of graphs, the authors recommend a number of difficulties and instructions for extra examine, and so they speak about interrelations of phrases and graphs within the literature by way of skill except word-representability.

The e-book is self-contained, and is acceptable for either reference and studying, with many chapters containing routines and suggestions to seleced difficulties. it is going to be priceless for researchers and graduate and complex undergraduate scholars in discrete arithmetic and theoretical desktop technology, specifically these engaged with graph conception and combinatorics, and in addition for experts in algebra.

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Additional resources for Words and Graphs

Example text

8. ([61]) The vertices of every (C4 , 2K2 , C5 )-free graph can be partitioned into a clique and an independent set. 8 are known as split graphs. 9. A graph is a split graph if and only if its vertices can be partitioned into a clique and an independent set. 8 shows that every (C4 , 2K2 , C5 )-free graph is a split graph. On the other hand, it is not difficult to see that none of the graphs C4 , 2K2 and C5 is a split graph, and moreover, each of them is a minimal non-split graph. Summarizing the above discussion we reach the following conclusion.

Then, if a word w represents G , then the word xxw represents G (in fact, xx can be inserted anywhere in w). Thus, when studying word-representability of a graph, we can ignore its isolated vertices, since it is trivial to represent them. 12. The reverse of the word w = w1 · · · wn is the word r(w) = wn wn−1 · · · w1 . 13. For the word w = 241533, the reverse r(w) = 335142. 5. 14. If w is a word-representant for a graph G, then the reverse r(w) is also (possibly the same) a word-representant for G.

Since vertices 2 and 3, and 1 and 2 are adjacent in Cn , we have 2i < 3i and 1i < 2i for each i. Then we have a contradiction with 1k < 2k < 3k < 1k . On the other hand, suppose that 1k+1 < 3k . Since all pairs of vertices j, j + 1, as well as the pair n, 1, are adjacent in Cn , for each i ≥ 1, we have that ji < (j + 1)i for each j < n and ni < 1i+1 . Thus, we obtain a contradiction with 3k < 4k < · · · < nk < 1k+1 < 3k . 11. Suppose that p(w) is the permutation obtained by removing all but the leftmost occurrence of each letter x in w.

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