arXiv subjects
Publications and source records attributed to Robert Lukoťka.
A rich $k$-flow is a nowhere-zero $k$-flow $ϕ$ such that, for every pair of adjacent edges $e$ and $f$, $|ϕ(e)| \neq |ϕ(f)|$. A graph is rich flow admissible if it admits a rich $k$-flow for some integer $k$. In this paper, we prove that if $G$ is a rich flow admissible graph with maximum degree $Δ$, then $G$ admits a rich $(264Δ- 445)$-flow.
A graph $G$ is cyclically $c$-edge-connected if there is no set of fewer than $c$ edges that disconnects $G$ into at least two cyclic components. We prove that if a $(k, g)$-cage $G$ has at most $2M(k, g) - g^2$ vertices, where $M(k, g)$ is the Moore bound, then $G$ is cyclically $(k - 2)g$-edge-connected, which equals the number of edges separating a $g$-cycle, and every cycle-separating $(k - 2)g$-edge-cut in $G$ separates a cycle of length $g$. In particular, this is true for unknown cages with $(k, g) \in \{(3, 13), (3, 14), (3, 15), (4, 9), (4, 10)$, $(4, 11),$ $(5, 7), (5, 9), (5, 10), (5, 11), (6, 7), (9, 7)\}$ and also the potential missing Moore graph with degree $57$ and diameter $2$. Keywords: cage, cyclic connectivity, girth, lower bound
A circular nowhere-zero $r$-flow on a bridgeless graph $G$ is an orientation of the edges and an assignment of real values from $[1, r-1]$ to the edges in such a way that the sum of incoming values equals the sum of outgoing values for every vertex. The circular flow number of $G$ is the infimum over all values $r$ such that $G$ admits a nowhere-zero $r$-flow. We prove that the circular glow number of Goldberg snark $G_{2k+1}$ is $4+1/(k+1)$, proving a conjecture of Goedgebeur, Mattiolo, and Mazzuoccolo.
A leaf matching operation on a graph consists of removing a vertex of degree~$1$ together with its neighbour from the graph. For $k\geq 0$, let $G$ be a $d$-regular cyclically $(d-1+2k)$-edge-connected graph of even order. We prove that for any given set $X$ of $d-1+k$ edges, there is no $1$-factor of $G$ avoiding $X$ if and only if either an isolated vertex can be obtained by a series of leaf matching operations in $G-X$, or $G-X$ has an independent set that contains more than half of the vertices of~$G$. To demonstrate how to check the conditions of the theorem we prove several statements on $2$-factors of cubic graphs. For $k\ge 3$, we prove that given a cubic cyclically $(4k-5)$-edge-connected graph $G$ and three paths of length $k$ such that the distance of any two of them is at least $8k-17$, there is a $2$-factor of $G$ that contains one of the paths . We provide a similar statement for two paths when $k=3$ and $k=4$. As a corollary we show that given a vertex $v$ in a cyclically $7$-edge-connected cubic graph, there is a $2$-factor such that $v$ is in a circuit of length greater than $7$.
A cycle cover of a graph is a collection of cycles such that each edge of the graph is contained in at least one of the cycles. The length of a cycle cover is the sum of all cycle lengths in the cover. We prove that every bridgeless cubic graph with $m$ edges has a cycle cover of length at most $212/135 \cdot m \ (\approx 1.570 m)$. Moreover, if the graph is cyclically $4$-edge-connected we obtain a cover of length at most $47/30 \cdot m \approx 1.567 m$.
Let $tsp(G)$ denote the length of a shortest travelling salesman tour in a graph $G$. We prove that for any $\varepsilon>0$, there exists a simple $2$-connected planar cubic graph $G_1$ such that $tsp(G_1)\ge (1.25-\varepsilon)\cdot|V(G_1)|$, a simple $2$-connected bipartite cubic graph $G_2$ such that $tsp(G_2)\ge (1.2-\varepsilon)\cdot|V(G_2)|$, and a simple $3$-connected cubic graph $G_3$ such that $tsp(G_3)\ge (1.125-\varepsilon)\cdot|V(G_3)|$.
We prove that every simple bridgeless cubic graph with n >= 8 vertices has a travelling salesman tour of length at most 1.3n - 2, which can be constructed in polynomial time.
We introduce weak oddness $ω_{\textrm w}$, a new measure of uncolourability of cubic graphs, defined as the least number of odd components in an even factor. For every bridgeless cubic graph $G$, $ρ(G)\leω_{\textrm w}(G)\leω(G)$, where $ρ(G)$ denotes the resistance of $G$ and $ω(G)$ denotes the oddness of $G$, so this new measure is an approximation of both oddness and resistance. We demonstrate that there are graphs $G$ satisfying $ρ(G) < ω_{\textrm w}(G) < ω(G)$, and that the difference between any two of those three measures can be arbitrarily large. The construction implies that if we replace a vertex of a cubic graph with a triangle, then its oddness can decrease by an arbitrarily large amount.
We show that every bridgeless cubic graph $G$ with $m$ edges has a cycle cover of length at most $1.6 m$. Moreover, if $G$ does not contain any intersecting circuits of length $5$, then $G$ has a cycle cover of length $212/135 \cdot m \approx 1.570 m$ and if $G$ contains no $5$-circuits, then it has a cycle cover of length at most $14/9 \cdot m \approx 1.556 m$. To prove our results, we show that each $2$-edge-connected cubic graph $G$ on $n$ vertices has a $2$-factor containing at most $n/10+f(G)$ circuits of length $5$, where the value of $f(G)$ only depends on the presence of several subgraphs arising from the Petersen graph. As a corollary we get that each $3$-edge-connected cubic graph on $n$ vertices has a $2$-factor containing at most $n/9$ circuits of length $5$ and each $4$-edge-connected cubic graph on $n$ vertices has a $2$-factor containing at most $n/10$ circuits of length $5$.
We show that every bridgeless cubic graph $G$ on $n$ vertices other than the Petersen graph has a 2-factor with at most $2(n-2)/15$ circuits of length $5$. An infinite family of graphs attains this bound. We also show that $G$ has a 2-factor with at most $n/5.8\overline{3}$ odd circuits. This improves the previously known bound of $n/5.41$ [Lukoťka, Máčajová, Mazák, Škoviera: Small snarks with large oddness, arXiv:1212.3641 [cs.DM] ].
A graph $G$ is $k$-degenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree $k$ or less. We prove that every connected planar graph with average degree $d \ge 2$ has a 4-degenerate induced subgraph containing at least $(38-d)/36$ of its vertices. This shows that every planar graph of order $n$ has a 4-degenerate induced subgraph of order more than $8/9 \cdot n$. We also consider a local variation of this problem and show that in every planar graph with at least 7 vertices, deleting a suitable vertex allows us to subsequently remove at least 6 more vertices of degree four or less.