Related papers: On Fan Raspaud Conjecture
A graph property is said to be elusive ( evasive) if every algorithm testing this property by asking questions of the form "is there an edge between vertices x and y" requires, in the worst case, to ask about all pairs of vertices. The…
A beautiful conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same…
Erd\H{o}s conjectured that every $n$-vertex triangle-free graph contains a subset of $\lfloor n/2\rfloor$ vertices that spans at most $n^2/50$ edges. Extending a recent result of Norin and Yepremyan, we confirm this conjecture for graphs…
Feder and Subi conjectured that for any $2$-coloring of the edges of the $n$-dimensional cube, we can find an antipodal pair of vertices connected by a path that changes color at most once. We discuss the case of random colorings, and we…
The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph and a matching. We show that this conjecture holds for the class of connected plane cubic graphs.
A {\em hole} in a graph is an induced subgraph which is a cycle of length at least four. A hole is called {\em even} if it has an even number of vertices. An {\em even-hole-free} graph is a graph with no even holes. A vertex of a graph is…
A \emph{queue layout} of a graph consists of a \emph{linear order} of its vertices and a partition of its edges into \emph{queues}, so that no two independent edges of the same queue are nested. The \emph{queue number} of a graph is the…
We call a finite undirected graph minimally k-matchable if it has at least k distinct perfect matchings but deleting any edge results in a graph which has not. An odd subdivision of some graph G is any graph obtained by replacing every edge…
A snark is a bridgeless cubic graph which is not 3-edge-colourable. The oddness of a bridgeless cubic graph is the minimum number of odd components in any 2-factor of the graph. Lukot'ka, M\'acajov\'a, Maz\'ak and \v{S}koviera showed in…
Let the class A of graphs be bridge-addable; that is, whenever a graph G in A has vertices u and v in different components then the graph G+uv is in A. For a random graph sampled uniformly from the graphs in A on vertex set {1,..,n}, there…
If K is an odd-dimensional flag closed manifold, flag generalized homology sphere or a more general flag weak pseudomanifold with sufficiently many vertices, then the maximal number of edges in K is achieved by the balanced join of cycles.…
A normal edge-coloring of a cubic graph is a proper edge-coloring, in which every edge is adjacent to edges colored with four distinct colors or to edges colored with two distinct colors. It is conjectured that $5$ colors suffice for a…
Visualizing a graph $G$ in the plane nicely, for example, without crossings, is unfortunately not always possible. To address this problem, Masa\v{r}\'ik and Hlin\v{e}n\'y [GD 2023] recently asked for each edge of $G$ to be drawn without…
A drawing of a graph is said to be a {\em straight-line drawing} if the vertices of $G$ are represented by distinct points in the plane and every edge is represented by a straight-line segment connecting the corresponding pair of vertices…
In this paper, we introduce the following new concept in graph drawing. Our task is to find a small collection of drawings such that they all together satisfy some property that is useful for graph visualization. We propose investigating a…
A 3-graph $\mathcal{F}$ is \emph{$U(s, 2s+1)$} if for any $s$ edges $e_1,...,e_s\in E(\mathcal{F})$, $|e_1\cup...\cup e_s|\leq 2s+1$. Frankl and Kupavskii (2020) proposed the following conjecture: For any $3$-graph $\mathcal{F}$ with $n$…
In this note, we prove that a finite vertex-transitive graph which has a clique which intersects all maximal cliques is a complete graph. This gives a positive answer in the case of vertex-transitive graphs to a question raised by Berge and…
There are many major open problems in integer flow theory, such as Tutte's 3-flow conjecture that every 4-edge-connected graph admits a nowhere-zero 3-flow, Jaeger et al.'s conjecture that every 5-edge-connected graph is $Z_3$-connected and…
We prove that $\{\overline{K_3}, H\}$-free graphs are not counterexamples to Hadwiger's Conjecture, where $H$ is any one of 33 graphs on seven, eight, or nine vertices, or $H=K_8$. This improves on past results of Plummer-Stiebitz-Toft,…
Here we prove that Reed Conjecture is valid for {P5, Flag_Complement}-free graphs where FlagComplement is the complement of the Flag graph. Some of the known results follow as corollaries to our result. Reed conjecture is still open in…