For a graph $H$ with $3\mid e(H)$, the zero-sum Ramsey number $R(H,\Z_3)$ is the least integer $N$ such that every labeling of the edges of $K_N$ by elements of $\Z_3$ contains a copy of $H$ whose edge labels sum to zero. We determine the last previously unresolved infinite family in the complete-graph case modulo $3$. More precisely, we prove that \(R(K_n,\Z_3)=n+3\) for every $n\ge 10$ satisfying $n\equiv 1\pmod 3$. Consequently, for $k\ge 1$, \(R(K_{9k+7},\Z_3)=9k+10\), resolving a problem of Caro and Mifsud.
Let $K_N^{(r)}$ denote the $N$-vertex complete $r$-uniform hypergraph. For an $r$-uniform hypergraph $H$ and an integer $k\geq2$, the $k$-color Ramsey number $R(H,k)$ is the least integer $N$ such that every $k$-edge-coloring of $K_N^{(r)}$ contains a monochromatic copy of $H$. When $k\mid\esize(H)$, the zero-sum Ramsey number $R(H,\mathbb Z_k)$ is the least integer $N$ such that every edge-labeling of $K_N^{(r)}$ by elements of $\mathbb Z_k$ contains a copy of $H$ whose edge labels sum to $0$ in $\mathbb Z_k$. We settle two conjectures and a problem concerning these two Ramsey numbers. First, Caro and Provstgaard proposed exact values for the zero-sum Ramsey numbers over $\mathbb Z_2$ of delta-systems with an even number of edges. We determine these numbers and thereby prove their conjecture. Second, for a forest $F$ with $m$ edges, let $tF$ denote the disjoint union of $t$ copies of $F$. Caro conjectured that $R(tF,\mathbb Z_{mt})=R(tF,2)$ for all sufficiently large $t$. We show that this conjecture does not hold for double stars. Caro also asked whether there exists a tree $T$ with $m$ edges such that $R(T,\mathbb Z_m)>R(T,2)$. We answer this question affirmatively by constructing an infinite family of such trees.
For graphs $G$ and $H$, let $\mathbf N(G,H)$ denote the number of unlabeled, not necessarily induced copies of $H$ in $G$, and let $\mathbf N_{\mathcal P}(n,H)$ be the maximum of $\mathbf N(G,H)$ over all $n$-vertex planar graphs $G$. We prove that, for every fixed integer $m\geq 3$, $$\mathbf N_{\mathcal P}(n,C_{2m+1})=2m\left(\frac{n}{m}\right)^m+O_m\!\left(n^{m-1/5}\right).$$ The proof uses a sharp weighted cycle--path inequality for edge probability measures on finite complete graphs. This strengthens a conjecture of Heath, Martin, and Wells and, together with their reduction lemma, yields the stated asymptotic formula.
For a simple graph $G$ of order $n$, let $S_2(G)=\lambda_1(G)+\lambda_2(G)$ denote its spectral sum. We determine, for every $n\geq5$, the exact maximum of $S_2(G)$ and all equality cases. The unique maximizer, up to isomorphism, is the complement of the disjoint union of a suitably balanced complete bipartite graph and isolated vertices, with the sizes of its three parts determined by $n$ modulo $7$. Denoting this graph by $K_n^\star$, we further show that $ S_2(K_n^\star)\leq\frac{8n}{7}-2,$ with equality exactly when $7\mid n$. This proves a conjecture of Kumar, Liu, Monterde, Pragada and Tait, which strengthens the Aouchiche--Hansen 2010 conjecture by extending it from connected graphs to all graphs and by asserting uniqueness of the extremal graph. The result also subsumes the 2008 conjecture of Ebrahimi B., Mohar, Nikiforov, and Ahmady. The proof combines Ky Fan's variational principle with a spectral inequality for weighted Ferrers quotients to reduce the problem to an explicit family whose complements have incidence rank one. Exact integer optimization and a separate equality analysis then yield the maximum and uniqueness.
For a graph $H$ and a family of graphs $\mathcal F$, let $\text{ex}(n,H,\mathcal F)$ denote the maximum number of copies of $H$ in an $\mathcal F$-free graph on $n$ vertices. For every integer $i\ge 3$, let $C_i$ denote the cycle of length $i$. For $r\ge 3$, set $\mathscr {C}_r=\{C_3,C_4,\ldots,C_r\},$ and set $\mathscr {C}_2=\varnothing$. In this paper, we prove that, for all integers $l>k\ge 2$, $$ \text{ex}(n,C_{2k+1},\mathscr {C}_{2k}\cup\{C_{2l+1}\}) =O_{k,l} \left(n^{2-\frac{1}{k(k+1)(l-k)}}\ \ \right). $$ Together with the known upper bounds for the number of triangles in $C_{2l+1}$-free graphs, this confirms a conjecture of Gerbner, Gy\H{o}ri, Methuku, and Vizer.
For a simple graph $G$ with $n$ vertices, write its chromatic polynomial in the rising factorial basis as $$ \chi_G(x)=\sum_{i=0}^{n}(-1)^{n-i}c_i(G)\langle x\rangle_i,$$ where $ \langle x\rangle_i=x(x+1)\cdots(x+i-1).$ The associated $\tau$-polynomial $$ \tau_G(x)=\sum_{i=0}^{n}c_i(G)x^i $$ was defined and systematically investigated by Brenti in 1992. In this paper, we prove that if the $\tau$-polynomials of two vertex-disjoint simple graphs $G$ and $H$ have only real zeros, then the $\tau$-polynomial of their join $G\vee H$ has only real zeros. This settles a conjecture posed by Brenti, Royle and Wagner since 1994.
For $1\le s<t$ and any graph $G$, the weakened Gallai-Ramsey number $gr^t_s(G)$ is defined to be the least $p\in \mathbb{N}$ such that every Gallai $t$-coloring of the edges of $K_p$ (i.e., a $t$-coloring that lacks rainbow triangles) contains a subgraph isomorphic to $G$ whose edges use at most $s$ of the colors. In the case of a book graph $B_n:=K_2+nK_1$, Jakhar and Moun determined the values $gr^3_2(B_3)=6$ and $gr^3_2(B_4)=7$. In this paper, we extend their results to $t>3$ colors, and we determine the values of $gr^3_2(B_n)$ for $5\le n\le 15$. General lower bounds for $gr^t_2(B_n)$ are also given.
Mark Budden, A. Gregory· 0 citations
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