No Arabic abstract
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 ErdH{o}s--Nev{s}etv{r}il conjecture regarding the strong chromatic index, Faudree et al. suggested investigating the strong clique number, and conjectured a quadratic upper bound in terms of the maximum degree. Recently, Cames van Batenburg, Kang, and Pirot conjectured a linear upper bound in terms of the maximum degree for graphs without even cycles. Namely, if $G$ is a $C_{2k}$-free graph, then $omega_S(G)leq (2k-1)Delta(G)-{2k-1choose 2}$, and if $G$ is a $C_{2k}$-free bipartite graph, then $omega_S(G)leq kDelta(G)-(k-1)$. We prove the second conjecture in a stronger form, by showing that forbidding all odd cycles is not necessary. To be precise, we show that a ${C_5, C_{2k}}$-free graph $G$ with $Delta(G)ge 1$ satisfies $omega_S(G)leq kDelta(G)-(k-1)$, when either $kgeq 4$ or $kin {2,3}$ and $G$ is also $C_3$-free. Regarding the first conjecture, we prove an upper bound that is off by the constant term. Namely, for $kgeq 3$, we prove that a $C_{2k}$-free graph $G$ with $Delta(G)ge 1$ satisfies $omega_S(G)leq (2k-1)Delta(G)+(2k-1)^2$. This improves some results of Cames van Batenburg, Kang, and Pirot.
Let $q_{min}(G)$ stand for the smallest eigenvalue of the signless Laplacian of a graph $G$ of order $n.$ This paper gives some results on the following extremal problem: How large can $q_minleft( Gright) $ be if $G$ is a graph of order $n,$ with no complete subgraph of order $r+1?$ It is shown that this problem is related to the well-known topic of making graphs bipartite. Using known classical results, several bounds on $q_{min}$ are obtained, thus extending previous work of Brandt for regular graphs. In addition, using graph blowups, a general asymptotic result about the maximum $q_{min}$ is established. As a supporting tool, the spectra of the Laplacian and the signless Laplacian of blowups of graphs are calculated.
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 $h(G)geq (1+o(1))ctsqrt{ln t}$ implies $G$ has a bipartite subgraph with Hadwiger number at least $t$, for some explicit $csim 1.276dotsc$. We improve this to $h(G) geq (1+o(1))tsqrt{log_2 t}$, and provide a construction showing this is tight. We also derive improved bounds for the topological minor variant of this problem.
We prove that the number of Hamilton cycles in the random graph G(n,p) is n!p^n(1+o(1))^n a.a.s., provided that pgeq (ln n+ln ln n+omega(1))/n. Furthermore, we prove the hitting-time version of this statement, showing that in the random graph process, the edge that creates a graph of minimum degree 2 creates (ln n/e)^n(1+o(1))^n Hamilton cycles a.a.s.
The edge-distinguishing chromatic number (EDCN) of a graph $G$ is the minimum positive integer $k$ such that there exists a vertex coloring $c:V(G)to{1,2,dotsc,k}$ whose induced edge labels ${c(u),c(v)}$ are distinct for all edges $uv$. Previous work has determined the EDCN of paths, cycles, and spider graphs with three legs. In this paper, we determine the EDCN of petal graphs with two petals and a loop, cycles with one chord, and spider graphs with four legs. These are achieved by graph embedding into looped complete graphs.
Let $G$ be a graph whose edges are coloured with $k$ colours, and $mathcal H=(H_1,dots , H_k)$ be a $k$-tuple of graphs. A monochromatic $mathcal H$-decomposition of $G$ is a partition of the edge set of $G$ such that each part is either a single edge or forms a monochromatic copy of $H_i$ in colour $i$, for some $1le ile k$. Let $phi_{k}(n,mathcal H)$ be the smallest number $phi$, such that, for every order-$n$ graph and every $k$-edge-colouring, there is a monochromatic $mathcal H$-decomposition with at most $phi$ elements. Extending the previous results of Liu and Sousa [Monochromatic $K_r$-decompositions of graphs, Journal of Graph Theory}, 76:89--100, 2014], we solve this problem when each graph in $mathcal H$ is a clique and $nge n_0(mathcal H)$ is sufficiently large.