Weakened Gallai-Ramsey Numbers for Books

Abstract: For $1\le s&lt;t$ 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 $$t\&gt;3$$ colors, and we determine the values of $gr<sup>3_2(B_n)$ for $5\le n\le 15$. General lower bounds for $gr<sup>t_2(B_n)$ are also given.

• The paper determines exact values of $gr^t_2(B_3)$ and $gr^t_2(B_4)$ for all $t\ge3$, including $6$ for $B_3$ and $7$ or $8$ for $B_4$ depending on the number of colors.
• The paper establishes exact values of $gr^3_2(B_n)$ for every $5\le n\le15$, ranging from $9$ for $B_5$ to $25$ for $B_{15}$, using Gallai structure, recursive block analysis, and classical Ramsey colorings.
• The paper derives residue-dependent lower bounds for all $t\ge3$, revealing roughly linear growth in $n$ and identifying open problems for larger books, more colors, and complete bipartite targets.

Background and motivation

A Gallai $t$-coloring of $K_p$ is an edge-coloring using at most $t$ colors that contains no rainbow triangle. The classical Gallai-Ramsey number $gr^t(G)$ is the least $p$ forcing a monochromatic copy of $G$ in every such coloring. This paper studies the weakened variant $gr^t_s(G)$, introduced in prior work by Beam–Budden and others: the least $p$ such that every Gallai $t$-coloring of $K_p$ contains a copy of $K_p$0 whose edges use at most $K_p$1 colors. The object of study is the book graph $K_p$2, with the $K_p$3 serving as the spine and the $K_p$4 isolated vertices as pages.

Prior to this work, Jakhar and Moun had determined $K_p$5 and $K_p$6. The paper extends these results to arbitrary numbers of colors for $K_p$7, determines exact values of $K_p$8 for all $K_p$9, and establishes general lower bounds valid for all $t$0. Since $t$1 whenever $t$2, well-definedness follows from the existence of ordinary Gallai-Ramsey numbers.

Structural tools

The upper-bound arguments rest on Gallai's structure theorem (in the Gyárfás–Simonyi formulation): every Gallai-colored complete graph arises by substituting Gallai-colored blocks into the vertices of a 2-colored base graph of order at least 2, with all edges between any two blocks sharing one color. Three lemmas streamline the case analysis:

• Large base graphs are forced: if the minimal base graph has order $t$3, then some pair of blocks has at most $t$4 vertices total, yielding a two-colored $t$5 directly.
• Base graphs of order 4: when $t$6, any Gallai 3-coloring whose base graph has order 4 contains a two-colored $t$7, proved by a counting argument on block sizes.
• Monochromatic vertex attachment: for $t$8 and $t$9, a coloring with a vertex incident only to red edges, whose remainder has minimal base graph of order 4, also forces a two-colored $gr^t(G)$0; the proof rules out alternative smaller base graphs via minimality contradictions.

Together these lemmas reduce most upper-bound proofs to base graphs of order 2, where pigeonhole and recursive peeling arguments apply.

General lower bounds

Two lower-bound constructions are given. The first is explicit and depends on $gr^t(G)$1: for $gr^t(G)$2 and $gr^t(G)$3,

$gr^t(G)$4

The constructions replace vertices of small 2-colored complete graphs ($gr^t(G)$5 or $gr^t(G)$6) with green cliques, possibly adding one green-dominated vertex, and verify that no color pair supports an $gr^t(G)$7-page book. The second construction is conditional: if there exists a 2-coloring of $gr^t(G)$8 avoiding both a monochromatic $gr^t(G)$9 and a monochromatic star $p$0, then $p$1. This leverages known critical colorings for 2-color book Ramsey numbers — notably Greenwood–Gleason's Paley graph of order 5 (for $p$2), Chvátal–Harary's Paley graph of order 9 (for $p$3), and Rousseau–Sheehan's Paley graph of order 13 (for $p$4).

Exact values for $p$5 and $p$6 with many colors

The paper proves $p$7 for all $p$8, extending Jakhar and Moun's $p$9 result. For $G$0, a qualitative distinction emerges:

$G$1	$G$2
$G$3	$G$4
$G$5	$G$6

The jump at $G$7 is witnessed by an explicit Gallai 4-coloring of $G$8 avoiding a two-colored $G$9; the matching upper bound on $gr^t_s(G)$0 follows from the structural reduction to order-2 base graphs and a double application of the pigeonhole principle. This dependence of $gr^t_s(G)$1 on $gr^t_s(G)$2 motivates restricting attention to $gr^t_s(G)$3 for larger books.

Exact values of $gr^t_s(G)$4 for $gr^t_s(G)$5

The main contribution is the following table of exact values:

$gr^t_s(G)$6	$gr^t_s(G)$7	Lower-bound source
5	9	general bound
6	11	critical coloring for $gr^t_s(G)$8
7	12	general bound
8	13	ad hoc construction
9	15	Chvátal–Harary coloring (subgraph)
10	17	general bound
11	19	Chvátal–Harary coloring
12	20	hybrid construction (orders 9 and 10)
13	22	general bound
14	23	ad hoc construction
15	25	Rousseau–Sheehan coloring (subgraph)

The upper bounds follow a uniform template. After the structural lemmas force a base graph of order 2 (or handle orders 2 and 4 separately), the argument splits into cases according to the block-size pair $gr^t_s(G)$9. Balanced blocks are dispatched by invoking known Ramsey numbers inside a block — e.g., $p$0, $p$1, $p$2, and $p$3 — so that a small two-colored book within one block combines with the other block as pages. Unbalanced blocks $p$4 recurse into the larger block, where the monochromatic-vertex lemma handles order-4 base graphs and further peeling handles order-2 ones. Notably, several proofs exploit the self-complementarity of Paley graphs, which makes their blue/green decompositions critical colorings for the corresponding book Ramsey numbers.

Two observations about the pattern of values deserve emphasis. First, the values track the lower bounds closely: for $p$5 the general bound is often tight (e.g., $p$6), while residues $p$7 tend to require the sharper Paley-based constructions. Second, the growth rate is roughly linear in $p$8 with slope between $p$9 and $t$0, consistent with the general lower bound; this contrasts with the exponential-in-$t$1 behavior of ordinary Gallai-Ramsey numbers for books established by Zou et al., reflecting the substantially weaker requirement that only two colors appear on the target subgraph.

Limitations and open questions

The authors are candid that the methods do not yet generalize. The upper-bound proofs depend heavily on tabulated values of $t$2 and $t$3, so extending the exact determination beyond $t$4 requires either new Ramsey data or genuinely new techniques; the case analysis also grows with $t$5. The behavior of $t$6 for $t$7 and $t$8 remains open, as does the relationship with complete bipartite targets: since $t$9, one has $K_p$0, with equality observed for $K_p$1, $K_p$2, $K_p$3, and additionally $K_p$4. Determining $K_p$5 for $K_p$6 is posed as a natural next step. Finally, the paper derives the monotonicity inequality $K_p$7 — a vertex outside the book can be absorbed without introducing more than one new color, else a rainbow triangle appears — which provides a starting point for the regime $K_p$8 but is not developed further here.

Conclusion

This paper substantially extends the theory of weakened Gallai-Ramsey numbers for books: it settles $K_p$9 and $K_p$00 for all $K_p$01, exhibits a color-dependence phenomenon at $K_p$02, supplies residue-class lower bounds valid uniformly in $K_p$03, and computes eleven new exact values of $K_p$04 through a systematic Gallai-partition framework augmented by classical critical colorings of Paley graphs. The remaining gap between the general lower bounds and exact values, and the extension to more colors or larger books, constitute the principal open problems left by this work.