Related papers: Embedding clique subdivisions via crux

The Hadwiger number $h(G)$ is the order of the largest complete minor in $G$. Does sufficient Hadwiger number imply a minor with additional properties? In [2], Geelen et al showed $h(G)\geq (1+o(1))ct\sqrt{\ln t}$ implies $G$ has a…

Let $G$ be a graph and $\mathcal{K}_G$ be the set of all cliques of $G$, then the clique graph of G denoted by $K(G)$ is the graph with vertex set $\mathcal{K}_G$ and two elements $Q_i,Q_j \in \mathcal{K}_G$ form an edge if and only if $Q_i…

Our main result is that every graph $G$ on $n\ge 10^4r^3$ vertices with minimum degree $\delta(G) \ge (1 - 1 / 10^4 r^{3/2} ) n$ has a fractional $K_r$-decomposition. Combining this result with recent work of Barber, K\"uhn, Lo and Osthus…

For a graph $G$, the central graph $C(G)$ is the graph constructed from $G$ by subdividing each edge of $G$ with one vertex and also by adding an edge to every pair of non-adjacent vertices in $G$. Also for a graph $G$, let $\gamma(G)$ and…

We prove that every $n$-vertex directed graph $G$ with the minimum outdegree $\delta^+(G) = d$ contains a subgraph $H$ satisfying \[ \min\left\{\delta^+(H), \delta^-(H) \right\} \ge \frac{d(d+1)}{2n} \,.\] We also show that if $d = o(n)$…

The Lescure-Meyniel conjecture is the analogue of Hadwiger's conjecture for the immersion order. It states that every graph $G$ contains the complete graph $K_{\chi(G)}$ as an immersion, and like its minor-order counterpart it is open even…

We first prove that for every vertex x of a 4-connected graph G there exists a subgraph H in G isomorphic to a subdivision of the complete graph K4 on four vertices such that G-V(H) is connected and contains x. This implies an affirmative…

Mader conjectured that every $C_4$-free graph has a subdivision of a clique of order linear in its average degree. We show that every $C_6$-free graph has such a subdivision of a large clique. We also prove the dense case of Mader's…

Reed conjectured that for any graph $G$, $\chi(G) \leq \lceil \frac{\omega(G)+\Delta(G)+1}{2}\rceil$, where $\chi(G)$, $\omega(G)$, and $\Delta(G)$ respectively denote the chromatic number, the clique number and the maximum degree of $G$.…

The quadratic embedding constant (QEC) of a graph $G$ is a new numeric invariant, which is defined in terms of the distance matrix and is denoted by $\mathrm{QEC}(G)$. By observing graph structure of the maximal cliques (clique graph), we…

Galinier, Habib, and Paul introduced the reduced clique graph of a chordal graph $G$. The nodes of the reduced clique graph are the maximal cliques of $G$, and two nodes are joined by an edge if and only if they form a non-disjoint…

A clique covering of a graph $G$ is a set of cliques of $G$ such that any edge of $G$ is contained in one of these cliques, and the weight of a clique covering is the sum of the sizes of the cliques in it. The sigma clique cover number…

For a clique cover $C$ in the undirected graph $G$, the {\it clique cover graph} of $C$ is the graph obtained by contracting the vertices of each clique in $C$ into a single vertex. The {\it clique cover width} of $G$, denoted by $CCW(G)$,…

A connected $t$-chromatic graph $G$ is \dfn{double-critical} if $G \backslash\{u, v\}$ is $(t-2)$-colorable for each edge $uv\in E(G)$. A long standing conjecture of Erd\H{o}s and Lov\'asz that the complete graphs are the only…

We consider the pattern detection problem in graphs: given a constant size pattern graph $H$ and a host graph $G$, determine whether $G$ contains a subgraph isomorphic to $H$. Our main results are: * We prove that if a pattern $H$ contains…

In 1990 Erd\H{o}s, Faudree, Rousseau and Schelp proved that for $k\geq 2$, every graph with $n\geq k+1$ vertices and $(k-1)(n-k+2)+\binom{k-2}{2}+1$ edges contains a subgraph of minimum degree $k$ on at most $n-\sqrt{n}/\sqrt{6k^3}$…

Given a graph $G$, the strong clique number of $G$, denoted $\omega_S(G)$, is the maximum size of a set $S$ of edges such that every pair of edges in $S$ has distance at most $2$ in the line graph of $G$. As a relaxation of the renowned…

Potential maximal cliques and minimal separators are combinatorial objects which were introduced and studied in the realm of minimal triangulations problems including Minimum Fill-in and Treewidth. We discover unexpected applications of…

We show that every minimally generically globally rigid graph in $\mathbb R^d$ which contains a subgraph isomorphic to $K_{d+2}$ is itself isomorphic to $K_{d+2}$, confirming a conjecture by Garamv{\"o}lgyi, Jackson, and Jord{\'a}n. The…

A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ contains a set $X$ of vertices such that $X$ meets all largest cliques of $H$, and $X$ induces a perfect graph. The chromatic number of a perfectly divisible graph $G$…