Graceful labellings of new families of windmill and snake graphs

A function ƒ is a graceful labelling of a graph G = (V,E) with m edges if ƒ is an injection ƒ : V (G) → {0, 1, 2, . . . ,m} such that each edge uv ∈ E is assigned the label |ƒ(u) − ƒ(v)| ∈ {1, 2, . . . ,m}, and no two edge labels are the same. If a graph G has a graceful labelling, we say that G itself is graceful. A variant is a near graceful labelling, which is similar, except the co-domain of f is {0, 1, 2, . . . ,m + 1} and the set of edge labels are either {1, 2, . . . ,m − 1,m} or {1, 2, . . . ,m − 1,m + 1}. In this thesis, we prove any Dutch windmill with three pendant triangles is (near) graceful, which settles Rosa’s conjecture for a new family of triangular cacti. Further, we introduce graceful and near graceful labellings of several families of windmills. In particular, we use Skolem-type sequences to prove (near) graceful labellings exist for windmills with C₃ and C₄ vanes, and infinite families of 3,5-windmills and 3,6-windmills. Furthermore, we offer a new solution showing that the graph obtained from the union of t 5-cycles with one vertex in common (Ct₅ ) is graceful if and only if t ≡ 0,3 (mod 4) and near graceful when t ≡ 1, 2 (mod 4). Also, we present a new sufficiency condition to obtain a graceful labelling for every kC₄ₙ snake and use this condition to label every such snake for n = 1, 2, . . . , 6. Then, we extend this result to cyclic snakes where the cycles lengths vary. Also, we obtain new results on the (near) graceful labelling of cyclic snakes based on cycles of lengths n = 6, 10, 14, completely solving the case n = 6

Decompositions of complete graphs into cycles of arbitrary lengths

We show that the complete graph on $n$ vertices can be decomposed into $t$ cycles of specified lengths $m_1,\ldots,m_t$ if and only if $n$ is odd, $3\leq m_i\leq n$ for $i=1,\ldots,t$, and $m_1+\cdots+m_t=\binom n2$. We also show that the complete graph on $n$ vertices can be decomposed into a perfect matching and $t$ cycles of specified lengths $m_1,\ldots,m_t$ if and only if $n$ is even, $3\leq m_i\leq n$ for $i=1,\ldots,t$, and $m_1+\ldots+m_t=\binom n2-\frac n2$.Comment: 182 pages, 0 figures, A condensed version of this paper was published as "Cycle decompositions V: Complete graphs into cycles of arbitrary lengths" (see reference [24]). Here, we include supplementary data and some proofs which were omitted from that pape

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum