Decomposition of the complete bipartite graph with a 1-factor removed into paths and stars

Let P_k denote a path on k vertices, and let S_k denote a star with k edges. For graphs F, G, and H, a decomposition of F is a set of edge-disjoint subgraphs of F whose union is F. A (G,H)-decomposition of F is a decomposition of F into copies of G and H using at least one of each. In this paper, necessary and sufficient conditions for the existence of the (P_{k+1},S_k)-decomposition of the complete bipartite graph with a 1-factor removed are given

Extending partial edge colorings of iterated cartesian products of cycles and paths

We consider the problem of extending partial edge colorings of iterated cartesian products of even cycles and paths, focusing on the case when the precolored edges satisfy either an Evans-type condition or is a matching. In particular, we prove that if $G=C^d_{2k}$ is the $d$th power of the cartesian product of the even cycle $C_{2k}$ with itself, and at most $2d-1$ edges of $G$ are precolored, then there is a proper $2d$-edge coloring of $G$ that agrees with the partial coloring. We show that the same conclusion holds, without restrictions on the number of precolored edges, if any two precolored edges are at distance at least $4$ from each other. For odd cycles of length at least $5$, we prove that if $G=C^d_{2k+1}$ is the $d$th power of the cartesian product of the odd cycle $C_{2k+1}$ with itself ($k\geq2$), and at most $2d$ edges of $G$ are precolored, then there is a proper $(2d+1)$-edge coloring of $G$ that agrees with the partial coloring. Our results generalize previous ones on precoloring extension of hypercubes [Journal of Graph Theory 95 (2020) 410--444]

Maximising the number of cycles in graphs with forbidden subgraphs

Fix $k \ge 2$ and let $H$ be a graph with $\chi(H) = k+1$ containing a critical edge. We show that for sufficiently large $n,$ the unique $n$-vertex $H$-free graph containing the maximum number of cycles is $T_k(n)$. This resolves both a question and a conjecture of Arman, Gunderson and Tsaturian \cite{Gund1}

Spotting Trees with Few Leaves

We show two results related to the Hamiltonicity and $k$-Path algorithms in undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10]. First, we demonstrate that the technique used can be generalized to finding some $k$-vertex tree with $l$ leaves in an $n$-vertex undirected graph in $O^*(1.657^k2^{l/2})$ time. It can be applied as a subroutine to solve the $k$-Internal Spanning Tree ($k$-IST) problem in $O^*(\min(3.455^k, 1.946^n))$ time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of $O^*(2^n)$. Second, we show that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for $k$-Path and Hamiltonicity in any graph of maximum degree $\Delta=4,\ldots,12$ or with vector chromatic number at most 8

Balanced Combinations of Solutions in Multi-Objective Optimization

For every list of integers x_1, ..., x_m there is some j such that x_1 + ... + x_j - x_{j+1} - ... - x_m \approx 0. So the list can be nearly balanced and for this we only need one alternation between addition and subtraction. But what if the x_i are k-dimensional integer vectors? Using results from topological degree theory we show that balancing is still possible, now with k alternations. This result is useful in multi-objective optimization, as it allows a polynomial-time computable balance of two alternatives with conflicting costs. The application to two multi-objective optimization problems yields the following results: - A randomized 1/2-approximation for multi-objective maximum asymmetric traveling salesman, which improves and simplifies the best known approximation for this problem. - A deterministic 1/2-approximation for multi-objective maximum weighted satisfiability

Forbidding induced even cycles in a graph: typical structure and counting

We determine, for all , the typical structure of graphs that do not contain an induced 2k-cycle. This verifies a conjecture of Balogh and Butterfield. Surprisingly, the typical structure of such graphs is richer than that encountered in related results. The approach we take also yields an approximate result on the typical structure of graphs without an induced 8-cycle or without an induced 10-cycle